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Vectors in Calculus

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
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- Integration of Logarithmic Functions
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- Intermediate Value Theorem
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- Jump Discontinuity
- Lagrange Error Bound
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- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
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- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
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- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
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- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
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- Rhombuses
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- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
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- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Power
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
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- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
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- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
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- Data Analysis
- Data Interpretation
- Degrees of Freedom
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- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
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- Hypothesis Test of Two Population Proportions
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- Inference for Distributions of Categorical Data
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If you asked for directions and simply got the answer "\(400\, m\)", this would not be helpful at all. What way should you go? Should you go \(400\, m\) left, right, forwards, backwards? For all you know, you might have to dig a hole \(400\, m\) deep underground, or fly \(400\, m\) into the air. This answer is not helpful because they have only provided a **scalar **quantity, a simple distance. If they had given you a distance with a direction, you would know exactly what to do. A distance with a direction is a **vector **quantity.

Vectors are mathematical objects representing movements or points in more than one dimension.

A **vector** is a mathematical object that has both **direction** and **magnitude**. A 2-dimensional vector can be written as:

\[ \begin{bmatrix} x \\ y\end{bmatrix} = x \vec{i} + y \vec{j}. \]

The left is called a **column vector**, and the second is called **component**** form.**

\( \vec{i}\) and \( \vec{j} \) are known as the** standard ****unit vectors**. These can be written as:

\[ \vec{i} = \begin{bmatrix} 1 \\ 0\end{bmatrix}, \: \vec{j} = \begin{bmatrix} 0\\1 \end{bmatrix}. \]

On a computer, vectors are often lowercase letters and written in bold. When handwritten, it is standard to underline them \((\underline{v})\), overline them \((\overline{v})\) or draw an arrow above them \((\overrightarrow{v})\). If you are specifically talking about a vector between two points, say point \(A\) and point \(B,\) this vector is normally written as the two points with an arrow above them, \( \overrightarrow{AB}.\)

Vectors can be thought of as arrows, pointing from one place to another. If the vector in 2d space is \( 3\vec{i} + 2 \vec{j},\) and the vector begins at the origin, it will point to \( (3, 2) \) on the \((x,y)\) plane.

The vector above could represent the movement of 3 units in the \(x\) direction and 2 units in the \(y\) direction, or it could represent the point \( (3, 2) \) in the \( (x,y)\) plane. Because of this, we distinguish vectors into **direction vectors **and **position vectors**.

The **direction vector **\( a \vec{i} + b \vec{j}\) is a vector representing a movement of \(a\) in the positive \(x\) direction and \(b\) in the positive \(y\) direction.

The **position vector** \( a \vec{i} + b \vec{j}.\) represents the point \( (a, b) \) in 2D space.

If you apply a direction vector from the origin, you will get to the corresponding position vector.

It is important to be able to write column vectors in component form and vice-versa. Let's look at some examples of this.

Vectors \(\vec{u}\) and \(\vec{v}\) are given below.

\[ \begin{align} \vec{u} & = \begin{bmatrix} 3 \\ -1 \end{bmatrix} \\ \vec{v} & = 3 \vec{i} + \vec{j}. \end{align} \]

Write

- Vector \(\vec{u}\) in component form,
- Vector \(\vec{v} \) in column vector form.

**Solution**

1. To write a column vector in component form, you write the number in the first position as the coefficient of \( \vec{i} \) and the number in the second position as the coefficient of \( \vec{j} \).

\[ \vec{u} = 3 \vec{v} - \vec{j}. \]

2. To write a vector in component form as a column vector, simply put the coefficients of each unit vector into their position in the column vector, remembering: the coefficient of \(\vec{i} \) goes in the first position, and the coefficient of \(\vec{j} \) goes in the second position. This gives:

\[ \vec{v} = \begin{bmatrix} 3 \\ 1 \\ -2 \end{bmatrix}. \]

Just as with normal numbers, vectors can be added, subtracted, and multiplied. Let's first look at the addition and subtraction of vectors.

When vectors are added, it is essentially like lining the arrows of the direction vectors tip to tip.

Above are two vectors \(\vec{v}\) and \(\vec{u}\), being added together. As you can see, the sum of these two vectors is the same as just stacking the vectors tip to tip. This makes sense when you think about a vector as a form of movement. If you first walk 3 steps right and 2 steps forwards, have a break, and then walk another step right and another two steps forward, you have moved to the same position as if you had just walked 4 steps forward and 4 steps right in one go.

Similarly to vector addition, a vector \(\vec{v}\) can be subtracted from a vector \(\vec{u}\) by putting them tip to tip, but with vector \(\vec{v}\) facing in the opposite direction.

Numerically, vectors can be added or subtracted by adding or subtracting the individual components. In component form, this makes visual sense, as it looks exactly the same as collecting like terms when doing algebra.

Find

- \( (\vec{i} + 2 \vec{j}) + (4\vec{i} - 3\vec{j}), \)
- \( ( 3\vec{i} - \vec{j}) - (\vec{i} - 2 \vec{j}). \)

**Solution**

1. You can add up all the terms as if the unit vectors were any other type of algebraic quantity,

\[ (\vec{i} + 2 \vec{j}) + (4\vec{i} - 3\vec{j}) = 5 \vec{i} - \vec{j}. \]

2. Here you can expand the bracket normally as if they were other algebraic quantities, and then simplify it:

\[ \begin{align} ( 3\vec{i} - \vec{j}) - (\vec{i} - 2 \vec{j}) & = 3 \vec{i} - \vec{j} - \vec{i} + 2 \vec{j}\\ & = 2 \vec{i} + \vec{j} \end{align} \]For column vectors, just add or subtract the numbers that are in the same position in each vector together.

Find

- \[ \begin{bmatrix} 3 \\ -1 \end{bmatrix} + \begin{bmatrix} 2 \\ -4 \end{bmatrix}. \]
- \[ \begin{bmatrix} 1 \\ -1 \end{bmatrix} - \begin{bmatrix} 4 \\ 1 \end{bmatrix}. \]

**Solution**

1. The number in the first position of our new vector will be the number in the first position of our first vector (3) added to the number in the first position of our second vector (2), so it will be 5. Do the same for the other row to get

\[ \begin{bmatrix} 3 \\ -1 \end{bmatrix} + \begin{bmatrix} 2 \\ -4 \end{bmatrix} = \begin{bmatrix} 5 \\ -5 \end{bmatrix}. \]

2. Do exactly the same as for question one, but this time subtracting the second number instead of adding it to the first corresponding number:

\[ \begin{bmatrix} 1 \\ -1 \end{bmatrix} - \begin{bmatrix} 4 \\ 1 \end{bmatrix} = \begin{bmatrix} -3 \\ -2 \end{bmatrix}. \]

There are many important properties of vectors within calculus, but first you must learn about the three methods of multiplication that exists for vectors.

**Vectors can be multiplied by real numbers**. The real numbers here are called **"scalers**", because they **scale the vector to a different size.**

If a vector is multiplied by 3, it is essentially the same as stacking 3 of that vector tip to tip. If the scalar is negative, the output of the multiplication will be facing in the opposite direction to the original vector. This reflects the multiplication of real numbers, as when you multiply \(x\) and \(y\) together, it is the same as adding \(x\) to itself \(y\) times.

Numerically, the** multiplication of a vector by a scalar** is done by **multiplying each component in the vector by that scalar**. For **a column vector**, this just means **multiplying each entry in the vector by the scalar.** For a vector in normal vector form, this looks just like expanding a bracket in any other equation.

This form of multiplication allows us to determine when two vectors are parallel.

A vector \(\vec{v}\) is parallel to another vector \(\vec{u}\) if and only if there is a scalar \(a\) such that \( \vec{v} = a \vec{u}.\)

Don't confuse the multiplication of a vector by a scalar with the scalar multiple of vectors. The scalar multiple of vectors is a way of multiplying vectors together, giving a scalar as the output. For more information, see Scalar Products.

Let's see some examples of multiplying a vector by a scalar.

The vectors \(\vec{v}\) and \(\vec{u}\) are given below,

\[ \begin{align} \vec{u} & = \begin{bmatrix} 2 \\ -1 \end{bmatrix} \\ \vec{v} & = 4 \vec{i} - \vec{j}. \end{align} \]

Find

- \[ 3 \vec{u}. \]
- \[ -\frac{1}{2} \vec{v}. \]

**Solution**

1. Multiply each component of the column vector by 3:

\[ \begin{align} 3 \vec{u} & = 3 \begin{bmatrix} 2 \\ -1 \end{bmatrix} \\ & = \begin{bmatrix} 6 \\ -3 \end{bmatrix}. \end{align} \]

2. Multiply each component of the column vector by \(-\frac{1}{2}\):

\[ \begin{align} -\frac{1}{2} \vec{v} & = -\frac{1}{2} (4\vec{i}-\vec{j}) \\&= -2 \vec{i}+\frac{1}{2}\vec{j} .\end{align} \]

The dot product and scaler product are two different ways of multiplying two vectors together. The dot product gives a scaler as the output, while the cross product gives another vector as an output.

Given two three dimensional vectors \( \vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k} \) and \( \vec{b} = b_1 \vec{i} + b_2 \vec{j} + b_3 \vec{k} \), the following are true:

The **dot product** or **scalar product** of a 2D vector is

\[ \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3. \]

The **cross product** or **vector product** of a 3D vector is

\[ \vec{a} \times \vec{b} = \begin{bmatrix} a_2 b_3 - a_3 b_2 \\ a_1 b_3 - a_3 b_1 \\ a_1 b_2 - a_2 b_1 \end{bmatrix}. \]

The dot product can be thought of as representing how much two vectors 'overlap'. This means that if two vectors are parallel and pointing in the same direction, the dot product will be maximized, but if the two vectors are orthogonal, the dot product will be 0.

The scalar product of two vectors gives a third vector that is perpendicular to both vectors, and is 0 when the vectors are perpendicular. In 2D, the dot product is considered the standard product, and the cross product does not exist. For AP, you do not need to work with vectors in more that 2 dimensions, so only the dot product is required.

Let's look at an example of finding the dot product

Given the following vectors:

\[ \vec{a} = \begin{bmatrix} 4 \\ -2 \\ 3 \end{bmatrix}, \hspace{1cm} \vec{b} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \]

find \( \vec{a} \cdot \vec{b}. \)

Multiply together the component pairs, and add them all up.

\[ \begin{align} \vec{a} \cdot \vec{b} & = 4 \cdot 1 + (-2) \cdot 0 + 3 \cdot (-1) \\ & = 4 + 0 -3 \\ & = 1 . \end{align} \]

Just as there are properties of regular arithmetic, such as associativity, distributivity and others, the same exist for vector arithmetic. For any vectors \(\vec{p}, \vec{q}, \vec{r}\) and scalars \( a, b\) the following properties hold:

**Commutativity**: \[ \vec{p} + \vec{q} = \vec{q} + \vec{p} .\]**Associativity of addition**: \[ (\vec{p} + \vec{q}) + \vec{r} = \vec{p} + (\vec{p} \vec{r}). \]**Distributivity of vectors**: \[ a (\vec{p} + \vec{q}) = a \vec{p} + a \vec{q} .\]**Distributivity of scalars**: \[ (a + b) \vec{p} = a \vec{p} + b \vec{p}. \]**Associativity of scalars**: \[ a (b \vec{p}) = (ab) \vec{p}. \]**Commutativity of dot product**: \[ \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}. \]**Distributivity of dot product over addition**: \[\vec{a} \cdot ( \vec{b} + \vec{c} ) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}. \]**Anticommutativity of cross product**: \[ \vec{a} \times \vec{b} = - \vec{b} \times \vec{a}. \]**Distributivity of cross product over addition**: \[ \vec{a} \times (\vec{b} + \vec{c} ) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}. \]**Scalar triple product**: \[\begin{align} \vec{a} \cdot ( \vec{b} \times \vec{c} ) &= \vec{b} \cdot ( \vec{c} \times \vec{a}) \\ &= \vec{c} \cdot (\vec{a} \times \vec{b}). \end{align}\]

There are many important formulas about vectors within mathematics. Here you will look at the most essential formulas.

An important equation in Vector mathematics is the **head minus tail rule**. This is a rule for calculating the vector between two points, if you have vectors going from another point to each of those two points.

Given vectors \(\vec{OA}, \vec{OB},\) the vector \(\vec{AB}\) is:\[ \vec{AB} = \vec{OB} - \vec{OA} .\]

If the point \(O\) is the origin, this simplifies to:

\[ \vec{AB} = \vec{B} = \vec{A}, \]

where \(\vec{A}, \vec{B}\) are the position vectors of points \(A\) and \(B\) respectively.

As stated in the definition of a vector, a vector has both direction and magnitude. Magnitude is the length of the vector, and can be calculated using the Pythagorean Theorem.

The magnitude of a vector is its length. For the vector,

\[ \vec{v} = \begin{bmatrix} x \\ y \end{bmatrix} = x \vec{i} + y \vec{j}, \]

the magnitude of \( \vec{v} \) is

\[ | \vec{v} | = \sqrt{ x^2 + y^2 }. \]

The magnitude of a vector can also be called the E**uclidean ****norm **(often just **norm**) or the **modulus** of the vector.

The magnitude of the vector represents the distance you would be travelling if you walked directly along the line. This distance is often known colloquially as "as the crow flies". Since this is a distance, the magnitude of a vector is always positive, assuming all of its components are real numbers.

The magnitude, or Euclidean norm, is not the only way of calculating the length of a vector, but it is the most common. In some scenarios, it makes more sense to use a different way of defining length.

For example, if you wish to get the Eurostar train from London to Amsterdam, you must first go from London to Brussels, and then from Brussels to Amsterdam. This is a much greater distance than the Euclidean distance from London to Amsterdam, so a different norm must be used when working out train travel distances.

The notation for the magnitude of a vector looks just like the absolute value of a real or complex number, and this is no coincidence. Just as the magnitude here represents the distance "as the crow flies" of the vector, the absolute value or modulus of a real or complex number is how far away that number is from the origin.

Let's look at some examples of calculating the magnitude of a vector.

Calculate the magnitude of the following vectors:

- \[ \vec{u} = 3 \vec{i} - 4 \vec{j} \]
- \[ \vec{v} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}. \]

**Solution**

1. You need to take the sum of the squares of the coefficients of the unit vectors, and then square root this answer. This will be

\[ \begin{align} | \vec{u} | & = \sqrt{3^2 + (-4)^2} \\ & = \sqrt{9 + 16} \\ & = \sqrt{25} \\ & = 5. \end{align} \]

2. This time, the components will be the values in the column vector. The magnitude of \( \vec{v} \) will be

\[ \begin{align} | \vec{v} | & = \sqrt{ 2^2 + (-4)^2} \\ & = \sqrt{ 4 + 16} \\ & = \sqrt{20}. \end{align} \]

In two dimensions, a vector can be determined using just the magnitude and the angle of the vector from the positive \(x\)-axis.

The formula for the angle \(\theta\) between a vector \(\vec{u} = x\vec{i} + y \vec{j}\) and the positive \(x\)-axis is:

\[\theta = \tan^{-1}{\frac{y}{x}}. \]

The angle should be between \(0^\circ\) and \(360^\circ,\) so you may have to add your answer to \(360^\circ\) if your calculator gives you a negative answer.

Let's look at an example of calculating the direction angle of a vector.

Find the direction angle of \(6 \vec{i} - 7 \vec{j}. \)

**Solution**

Using the formula, with \(6\) in place of \(x\) and \(-7\) in place of \(y\) gets:

\[ \theta = \tan^{-1}{\frac{-7}{6}} = -49.40^\circ \]

to 2 decimal places. This angle is not between \(0^\circ\) and \(360^\circ,\) so you must add \(360^\circ\) to it:

\[ \theta = -49.40^\circ + 360^\circ = 310.80^\circ\]

to 2 decimal places. This is the size of the direction angle.

An important type of vector in mathematics is the **unit vector. **You have already met some unit vectors in this article, the **standard unit vectors**, \( \vec{i}\) and \( \vec{j}.\)

A **unit vector** is a vector with a magnitude equal to \(1\).

The **normalized vector** of a vector \( \vec{v} \) is the unit vector pointing in the same direction as vector \( \vec{v}. \) This is denoted by \( \hat{v} \), and is often referred to as "v hat". \( \hat{v} \) can be calculated by:

\[ \hat{v} = \frac{1}{|\vec{v}|} \vec{v}. \]

Let's look at normalizing some vectors.

Normalize the following vectors:

- \[\vec{v} = \begin{bmatrix} 3 \\ -4 \end{bmatrix} \]
- \[\vec{u} = 5 \vec{i} -2 \vec{j}. \]

**Solution**

1. First, calculate the magnitude of the vector:

\[ \begin{align} | \vec{v} | & = \left| \begin{bmatrix} 3 \\ -4 \end{bmatrix} \right| \\ & = \sqrt{3^2 + (-4)^2 } \\ & = \sqrt{9 + 16 + } \\ & = \sqrt{25} = 5. \end{align} \]

Now, multiply the vector by \( \frac{1}{|\vec{v}|} \) to get \( \hat{v} \). Remember that to multiply a vector by a scalar, you multiply each of the entries in the vector by the scalar. This gives you

\[ \begin{align} \hat{v} & = \frac{1}{|\vec{v}|} \vec{v} \\ & = \frac{1}{5} \begin{bmatrix} 3 \\ -4 \end{bmatrix} \\ & = \begin{bmatrix} \frac{3}{5} \\ \frac{-4}{5} \\ \frac{1}{5} \end{bmatrix} \end{align}\]

This is the final normalized vector.

2. Again, the first step is to calculate the magnitude of the vector:

\[ \begin{align} | \vec{v} | & = | 5 \vec{i} -2 \vec{j}| \\ & = \sqrt{5^2 + (-2)^2} \\ & = \sqrt{25 + 4} \\ & = \sqrt{29}. \end{align} \]Now, multiply \(\vec{u} \) by \( \frac{1}{| \vec{u} |} \). Since it is in unit vector form, this is just like expanding the brackets:

\[ \begin{align} \hat{u} & = \frac{1}{|\vec{u}|} \vec{u} \\ & = \frac{1}{\sqrt{29}} (5 \vec{i} -2 \vec{j}) \\ & = \frac{5}{\sqrt{29}} \vec{i} -\frac{2}{\sqrt{29}} \vec{j}. \end{align} \]

This is the normalized vector.

The cross and dot product both have formulas that allow you to calculate the angle \(\theta\) between two vectors.

Given two vectors \(\vec{a}\) and \(\vec{b},\) these formulas are:

\[ \begin{align} \vec{a} \cdot \vec{b} &= | \vec{a} | | \vec{b} | \cos{\theta} \\ | \vec{a} \times \vec{b} | &= |\vec{a} | | \vec{b} | \sin{\theta}. \end{align} \]

Vector-valued functions are types of functions that take a scalar \(t\) as input, and output a vector. In physics, it is standard to define position as a vector-valued function \(\vec{s}(t),\) where the input \(t\) is time and the output is the position vector at time \(t.\) This makes for a useful way of defining the way that projectiles move in 2D or 3D space.

The velocity of the particle can then be calculated by differentiating the given position function \( \vec{s}(t), \) and the acceleration will be the second derivative of the position function, and hence the derivative of the velocity function. For more information, see Vector-Valued Functions and Vector-valued motion - position, speed, acceleration.

Many areas of Physics require a good knowledge of vector calculus, including mechanics, quantum physics and general relativity. Beyond physics, vector calculus is also an essential part of modern computer programming, including graphics and machine learning. In fact, gradient descent, an incredibly important part of machine learning, is the method of calculating the steepest descent to find the local minimum of a function. Within machine learning, this is essential as it allows for the minimization of loss functions or error within the algorithm. For more applications see Vector-valued motion - position, speed, acceleration.

- A
**vector**is a mathematical object that has both direction and magnitude, and can represent points (known as**position vectors**) or movements (known as**direction vectors**). - Vectors can be written in
**column vector form**: \( \begin{bmatrix} x \\ y\end{bmatrix}, \) or**component****form**: \( x \vec{i} + y \vec{j}. \) These definitions are equivalent. - Vectors can be
**added**or subtracted by**adding**or subtracting the individual components together. In column vector form, just add together each row. In unit vector form, collect like terms. - Vectors can be multiplied by scalars. The output of multiplying by scalar \(a\) is a vector that is parallel to the original vector but \(a\) times longer and facing in the opposite direction if \(a\) is negative.
- The
**magnitude**of a vector is the length of the vector. if the vector is \( \vec{v} = \begin{bmatrix} x \\ y \end{bmatrix} \), then the magnitude of the vector is: \[ | \vec{v} | = \sqrt{ x^2 + y^2}. \] - A
**unit vector**is a vector with magnitude 1. The**normalized vector**of the vector \(\vec{v}\) is the unit vector pointing in the same direction as \( \vec{v}. \) The**normalized vector**of \( \vec{v} \) is: \[ \hat{v} = \frac{1}{|\vec{v}|} \vec{v}. \] - The
**dot product**or**scalar product**of is\[ \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 = | \vec{a} | | \vec{b} | \cos{\theta} . \]

A vector is a mathematical object that has both a direction and a magnitude, or size.

A unit vector is a vector that has a magnitude or length of 1.

The angle between two vectors can be calculated using the following formula:

cos^-1 ((**a** . **b**)/(|**a**| |**b**|)).

A vector valued function is a function that takes a scalar value as input and outputs a vector.

More about Vectors in Calculus

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