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Vectors

- 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
- 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
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- 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
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- 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
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- 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
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- 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
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- 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
- 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
- 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
- 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
- Binomial Hypothesis Test
- 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
- Conducting a Survey
- 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
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

A physical quantity is something that one can measure. When we talk about quantities, we either describe them as a scalar quantity or a vector quantity.

A **scalar quantity **is one that only has magnitude. For example, speed is a scalar quantity as it only describes how quickly something is moving, not the direction in which it moves.

A **vector quantity **has both magnitude and direction. An example is velocity, as here we must give the speed an object is moving at and in which direction (eg 20kph due north).

Now we know what a vector quantity is, how does that relate to maths? Let's first think about how a vector would look in two dimensions. Usually, in two dimensions, we describe things using an x-coordinate and a y-coordinate, as this can describe any point in the xy-plane. By doing this, we describe how far we move in the x direction and then how far we move in the y direction, meaning we have satisfied our need for magnitude and direction.

We can write this vector in a couple of ways. The most common way is as a column vector. In 2D, this looks like , where x corresponds to the x coordinate and y to the y coordinate. We can also write a vector using unit vectors. In 2D, our unit vectors are represented by ** i** and

By definition, we take a positive x value in our vector to mean going to the right of the origin. Conversely, a negative x value represents going to the left of the origin. Again, by definition, we take a positive y value to mean pointing upwards from the origin, and conversely, a negative y value represents going down from the origin.

For example, if I want to show the vector on a coordinate axis, it would be as shown below.

Notice here that to show the vector , I have shown the vector first and then the vector . It is important to note that whenever we are drawing vectors, we must draw on an arrow to show the direction of the vector. Without this, we wouldn't be able to identify the vector from the vector . When we write a vector going from, eg point A to point B, we denote this vector . If it were to go the other way, it would be called . Note that the arrow shows the direction of travel.

We can extend this thinking to three dimensions. In the z-direction, the unit vector is denoted by * k* . This looks like in column vector form. A three-dimensional axis looks like this:

The thing to remember is that all three axes are perpendicular to each other. (If your tabletop were the xy plane, the z-direction would be coming directly out from the table.) If we had a vector of the form **r** = , we could express it as **r** = x ** i** + y

When looking at vectors, we can also directly describe their magnitude and direction. However, it is not obvious how to relate what we see geometrically to the ideas of magnitude and direction. We will explore this further now.

Visually, the magnitude of a vector is its length. We can show the magnitude of vector **a** as | **a** |.

Let's first think about this in two dimensions, and then we can extend these ideas to three dimensions. In 2D, we can draw a right-angled triangle around our vector, using the vector itself as the hypotenuse of the triangle. This Means we can use Pythagoras' Theorem to find the formula for a vector in 2D: .

A sketch of the vector and the triangle are shown below.

We now need to extend this idea to three dimensions. As before, we need to add the z term to the 2D formula, and so the formula in three dimensions is

Find the magnitude of the vector

Using the formula for magnitude, we get

Find the magnitude of the vector .

Using the formula for magnitude, we get

We measure the direction of a vector with respect to another point. In two dimensions, this is usually the x-axis, but we always need to specify. Let's draw a diagram to explain this better.

We give directions in terms of , so this means that in two dimensions, the direction of any vectorwith respect to the x-axis is given as. If we wanted to measure the angle with respect to the y-axis, this would be given as . Notationally, we take anything above the x-axis as a positive value of theta, and anything below the axis will give a negative value of theta.

We can extend this to any axis in three dimensions. Suppose we have a vector , and we want to find the angle with each axis. The formulas are given as follows:

x-axis:

y-axis:

z-axis:

Find the direction of **a ** = 3 ** i** - 2

First, let's work out | **a** | : . Then by the formula, the angle with the x-axis is given by , which gives .

Some calculations require you to add and subtract vectors.

How do we add two vector quantities together? Let's first think of this geometrically. To add two vectors together, we would draw our first vector as usual, starting from the origin. We would then draw the second vector starting at the end of our first vector. We would then draw a line from the origin to the end of the second vector to get the result of adding the two vectors. This new vector is called the **r****esultant vector** . This process is shown below.

Now we know how to do this geometrically. What is the process algebraically? It turns out that we can add two vectors simply by adding the corresponding components. This means that the formula for adding two vectors is given by in two dimensions. The process is exactly the same in three dimensions.

Add the vectors and .

Using the formula, we get

Add the vectors and

We can add components to get

This is very similar both geometrically and algebraically. Geometrically, when we take away a vector, we reverse its direction and then follow the same process. Algebraically, we take away components, so for two vectors, we get

. Again, the process is exactly the same in three dimensions.

Calculate

Using the given formula, we get

Calculate

Subtraction elementwise gives us

When we talk about multiplying a vector by a scalar, it looks like this: . Geometrically, we can see this as the vector joined to itself over and over a times. This means that for any non-zero a,is parallel to. Algebraically, this means that. Again, we can follow the process for three dimensions.

Calculate

Using the formula, we get this as

Suppose we are given two points in vector form and , and we wish to find the direct distance between these points. We could find a vector connecting these two points by taking one vector away from the other. We have the distance between the points as a vector, so we now need to find the length of the vector. This is given by the magnitude of this vector.

Find the distance between and .

First we need to find the vector connecting and . We do this by taking away one vector from the other . We now need to find the magnitude of this vector. This is given by - which is our answer.

You might be asked about geometric proof with vectors. This type of question will typically look like a shape annotated with vector lengths. It will ask us to show something - the length of a certain side as a combination of the given lengths, proving two sides are parallel or perpendicular, or proving midpoints. These are best shown through examples.

P is a point on AB such that AP: PB = 1: 3

Show that is

To find we can break this down to .

, and we can break down as .

This means that .

Thus,

C is a point on AO such that AC: CO = 3: 1

M and N are the midpoints of BC and BO respectively

Let us denoteby **a** and by **b.**

Show that MN is parallel to AO.

, so we need to show MN is k **a** for any constant k.

, by breaking it down.

.This means we can write , which satisfies what we needed to show. QED

Now we've seen the mathematical interpretation of vectors, how do we apply these to real life? We've already discussed velocity being a vector quantity. When we describe velocity, we give the magnitude (commonly known as speed in this context) and the direction of travel. Position, velocity and acceleration are all vector quantities related to each other (velocity is the rate of change of position, and acceleration is the rate of change of velocity).

Forces are also a vector quantity, as the direction and magnitude of the force matter. This means that quantities such as thrust, weight and momentum are also vector quantities.

We can apply our knowledge of vectors to other types of questions, too. Examples would include finding the position vector of a missing vertex of a shape or translating a shape by a vector. Let's look at how to calculate these.

Translate the rectangle given by (0,0), (4,7), (4,0), (0,7) by the vector .

Visually, this looks like moving the shape one unit in the positive x direction and four units in the positive y direction. To find the coordinates of the translated rectangle, we would need to add the vector to each position vector of a corner, which would give us (1, 4), (5, 11), (5, 4 ), (1, 11).

Note: We switch between coordinates and position vectors freely here, as they represent the same thing for our purposes.

A parallelogram has given vertices . Find the missing vertex.

The best place to start here is by plotting points. This gives us

For the completed shape to be a parallelogram, we need the y coordinate to be 4. Then we can find the x coordinate by finding the difference of x between (1, 1) and (8, 1) and adding it to the x value of the top point (shown in black). This gives us 7, meaning that the x coordinate is 10, giving the final point as .

,*i*,**j**are the unit vectors in the x, y and z directions, respectively.**k**A vector has direction and magnitude.

The magnitude of a vector is given by the square root of the sum of the components of the vector squared.

Vector addition, subtraction and scalar multiplication occur elementwise.

A vector is a quantity in which we can express both direction and magnitude.

To find the magnitude of any vector, calculate the square root of the sum of the squared components.

To find out the magnitude of a vector x, divide the components of the vector by its magnitude.

More about Vectors

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