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Vector Valued Function

Vector Valued Function

When working with objects moving through space, it makes sense to consider them moving over a certain amount of time, \(t.\) Time could be drawn as another dimension on a graph, but most of the time this is unnecessary since time always carries on in the same way (assuming your not dealing with anything traveling near the speed of light.) For this reason, it is often useful to define the position on the \(x\) and \(y\) axis using time, but not writing time as a third axis. This is something that does not work so well with Cartesian equations but is much simpler using vector-valued functions, hence making them incredibly useful in Physics, Machine Learning, and many other subjects.

Vector-Valued Function Definition

Before you get into the details of vector-valued functions, it is important to understand vectors fully.

Vectors

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

Vector-Values Functions A vector represented by an arrow StudySmarterA vector can be thought of like an arrow, pointing from one place to another.

Vectors can be written in two different ways,

  • Column Vector Form: \( \begin{bmatrix} x \\ y \end{bmatrix}, \)

  • Component Form: \( x \vec{i} + y \vec{j}. \)

These two vectors are equivalent. Numerically, vectors can be added and subtracted by adding or subtracting the individual components. Similarly, they can be multiplied by scalar quantities by multiplying the individual components. In component form, this looks just like collecting like terms and expanding brackets.

Graphically, adding vectors is done by stacking them tip to tip, and subtracting by stacking them tip to tip, but pointing the second vector in the opposite direction. Multiplying numbers by a scalar \(\lambda\) is the same as stacking \(\lambda\) of the same vectors, tip to tip, and if \(\lambda\) is negative, the product will be pointing in the opposite direction.

Finally, given a vector \( v = x \vec{i} + y \vec{j},\) the magnitude \(|\vec{v}|\) and direction angle \(\theta\) of a vector can be calculated using the following formulas:

\[ \begin{align} | \vec{v} | & = \sqrt{ x^2 + y^2 }, \\ \theta & = \tan^{-1}\left({\frac{x}{y}}\right) \end{align} \]

For more information on all of this, see Vectors.

What are Vector-Valued Functions?

Vector-valued functions are just like real-valued functions, but output a vector instead of a scalar.

A vector-valued function is a function that takes a scalar value as input, and gives a vector as output. A vector-valued function of one variable looks like this,

\[ \vec{r}(t) = \begin{bmatrix} f(t) \\ g(t) \end{bmatrix} = f(t) \vec{i} + g(t) \vec{j}. \]

Here, \( f(t)\) and \(g(t)\) are parametric equations.

Given this definition, you can deduce the domain and range of a vector-valued function.

  • The domain of a vector-valued function is a subset of \(\mathbb{R},\)

  • The range of an \(n\)-dimensional vector-valued function is a subset of \(\mathbb{R}^n.\)

Here you will focus on vectors in 2 dimensions, meaning the range of the functions will be a subset of \(\mathbb{R}^2.\) It is important to note that it is a subset of \(\mathbb{R}^2\) and not the whole of \(\mathbb{R}^2,\) since you will encounter many vector-valued functions that cannot output to every point in \(\mathbb{R}^2.\)

Examples of Vector-Valued Functions

There are many different types of vector-valued functions, but here you will look at some of the simplest.

Straight Lines

The vector-valued formula for a straight line is

\[ \vec{r}(t) = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} + t \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}. \]

Here, \(\vec{a} = a_1 \vec{i} + a_2 \vec{b} \) is the position vector of a point \(a\) on the line, and \( \vec{b} = b_1 \vec{i} + b_2 \vec{j} \) is a vector that is parallel to the line.

Vector-valued functions A line with a point on it labelled a, and a parallel vector labeled b StudySmarterA line is defined by a vector-valued function using a point on the line, \(a,\) and a vector parallel to the line, \(\vec{b}.\)

Circles and Ellipses

The vector-valued equation for a circle with radius \(a\) is

\[ \vec{r}(t) = \begin{bmatrix} a \cos{t} \\ a \sin{t} \end{bmatrix} \]

Vector-valued functions a circle with radius a, and points on the curve labelled a cosine t, a sine t StudySmarterThe vector-valued function for a circle can be made using the sine and cosine functions.

An ellipse can be defined similarly, but using \(a\) as the intercept on the \(x\)-axis and \(b\) as the intercept on the \(y\)-axis.

\[ \vec{r}(t) = \begin{bmatrix} a \cos{t} \\ b \sin{t} \end{bmatrix} \]

Vector-valued functions A circle with intercepts a and b on the x and y axis, with lengths a cosine t and b sine t going from each axis to the ellipse StudySmarterThe vector-valued function for an ellipse can be defined similarly to that of a circle, but taking into account the different axis intercepts.

Spirals

There are many ways to define spirals in mathematics, but an easy way is to define them similarly to spirals and circles, but with a \(t\) term in front of the trigonometric functions.

\[ \vec{r}(t) = \begin{bmatrix} a t \cos{t} \\ b t \sin{t} \end{bmatrix} \]

Vector-valued Functions A line spiraling outwards from the origin StudySmarterThe graph of an spiral, where \(a = b = \frac{1}{2}. \)

Graphing Vector-Valued Functions

When you first learnt to graph Cartesian equations such as \(y = f(x),\) you likely started by drawing a table of values for \(x,\) and then filling in the corresponding values of \(y.\) You could then plot these points and join them up, to create an estimation of the curve. You can do the exact same thing to graph vector-valued functions, but instead starting with the variable \(t\) and using these values of \(t\) to calculate the corresponding values of \(x\) and \(y.\) Let's look at an example of this.

Sketch the graph of \( \vec{r} = t^2 \vec{i} + t \vec{j}, \) for values of \(-4 < t < 4. \)

Solution

First, create a table with three columns, titled \(t, x, y.\) You can fill in the \(t\) column with the integers from \(-4\) to \(4.\)

\(t\)
\(x\)
\(y\)
-4
-3
-2
-1
0
1
2
3
4

From here, you can start filling in the values. Remember that \(x\) will be the coefficient of the \(\vec{i}\) term, and \(y\) will be the coefficient of the \(\vec{j}\) term. First, let's fill in the \(x\) column by squaring all of the values in the \(t\) column.

\(t\)
\(x\)
\(y\)
-4
16
-3
9
-2
4
-1
1
0
0
1
1
2
4
3
9
4
16

Next, fill in the \(y\) column. This will be exactly the same as the values on the \(t\) column.

\(t\)
\(x\)
\(y\)
-4
16
-4
-3
9
-3
-2
4
-2
-1
1
-1
0
0
0
1
1
1
2
4
2
3
9
3
4
16
4

Next, plot the \((x,y)\) pairs on a graph.

Vector-valued functions the points make the shape of a parabola going in the positive x direction StudySmarterThe shape of these dots seems to resemble a parabola.

Based on the shape of the plotted points and the fact that the function has a \(t^2\) term in it, it appears to be a parabola. You can draw a curve between these points to get the following curve:

Vector-valued functions The parabola x = y squared, for values of x between -4 and 4 StudySmarterThe finished curve is the parabola \(x = y^2.\)

To see more examples, see Graphing Vector-Valued Functions.

Vector-Valued Functions Formula

The most important formula for vector-valued functions is the formula for arc length, or the length of a curve between two points.

Vector-valued Functions a curve with the length between t = a and t = b highlighted StudySmarterThe length of the curve between the points \(t=a\) and \(t=b.\)

The length \(L\) of a curve \(\vec{r}(t) = f(t) \vec{i} + g(t) \vec{j} \) between two point \(a\) and \(b\) is

\[ L = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2} \, \mathrm{d}t. \]

This measures the whole length of the curve as if you had laid a piece of string on the curve and then cut it off and measured it. Let's look at some examples using this formula.

Find the arc length of

\[ \vec{r} = \begin{bmatrix} \sin{(3t)} \\ \cos{(3t)} \end{bmatrix} \]

for \(-4 < t < 2.\)

Solution

Here, \(f(t) = \sin{(3t)}\) and \(g(t) = \cos{(3t)}.\) The formula requires the derivatives of these functions, so you must differentiate them both.

\[ \begin{align} f'(t) & = 3 \cos{(3t)} \\ g'(t) & = 3 \sin{(3t)}. \end{align} \]

From here, you can substitute these into the formula for the arc length.

\[ \begin{align} L & = \int_{-4}^{2} \sqrt{(3 \cos{(3t)})^2 + (3 \sin{(3t)})^2} \, \mathrm{d}t \\ & = \int_{-4}^2 \sqrt{ 9 \cos^2{(3t)} + 9 \sin^2{(3t)} } \, \mathrm{d}t \\ & = \int_{-4}^{2} \sqrt{9 (\cos^2{(3t)} + \sin^2{(3t)})} \, \mathrm{d}t. \end{align} \]

From here, you can use the formula \(\sin^2{x} + \cos^2{x} = 1. \)

\[ \begin{align} L & = \int_{-4}^{2} \sqrt{9 \cdot 1} \, \mathrm{d}t \\ & = \int_{-4}^{2} 3 \, \mathrm{d}t \\ & = [3t]_{-4}^{2} \\ & = 3\cdot 2 - 3 \cdot (-4) \\ & = 18. \end{align} \]

Hence, the arc length is 18 unit length.

Derivatives of Vector-Valued Function

The derivative of vector-valued functions can be found by differentiating each component of the vector-valued function. The derivative of \( \vec{r}(t) = f(t) \vec{i} + g(t) \vec{j} \) is:

\[ \frac{\mathrm{d}\vec{r}}{\mathrm{d}t}(t) = \frac{\mathrm{d}f}{\mathrm{d}t}(t) \vec{i} + \frac{\mathrm{d}g}{\mathrm{d}\mathrm{d}}(t) \vec{j}, \]

assuming that the derivatives of \(f(t)\) and \(g(t)\) with respect to \(t\) exist. This makes sense logically, as it is just like using the addition rule when differentiating any other function. The derivative of a vector-valued function at a point will point in the direction of travel of the function, at a tangent to the curve.

If the vector valued function, call it \(\vec{s}(t),\) represents position on the \(xy\) plane at time \(t,\) then the derivative of this function will be the velocity vector \(\vec{v}(t).\) The magnitude of the velocity vector at time \(t\) is the speed of travel at time \(t.\) Similarly, the differential of the velocity vector will be the acceleration vector, \( \vec{a}(t). \) Let's take a look at differentiating some vector-valued functions.

A particle's position in space is given by the vector-valued function

\[ \vec{s}(t) = \begin{bmatrix} 3t^2 \\ e^t \end{bmatrix}. \]

Find the vector-valued functions for the velocity and acceleration of the particle.

Solution

If you differentiate the position function, you will get the velocity function. This will be,

\[ \vec{v}(t) = \vec{s}'(t) = \begin{bmatrix} 6t \\ e^t \end{bmatrix}. \]

Next, you can differentiate this again to find the acceleration function.

\[ \vec{a}(t) = \vec{v}'(t) = \begin{bmatrix} 6 \\ e^t \end{bmatrix}. \]

To learn more about differentiating vector-valued functions, see Calculus of Vector-Valued Functions.

Vector Valued Function - Key takeaways

  • A vector-valued function is a function that takes a scalar value as input, and gives a vector as an output.
  • Vector-valued functions can be written \( \vec{r}(t) = f(t) \vec{i} + g(t) \vec{j}. \)
  • The domain of a vector-valued function is a subset of \(\mathbb{R}\).

  • The range of an \(n\)-dimensional vector-valued function is a subset of \(\mathbb{R}^n.\)

Frequently Asked Questions about Vector Valued Function

vector valued functions can be written as:

r(t) = f(t) i + g(t) j,

where f(t) and g(t) are scalar functions.

A vector valued function written as:
r(t) = f(t) i + g(t) j.

The components of this are:

  • The unit vectors i and j,
  • The scalar functions f(t) and g(t).

A vector valued function will have the range of Rn, where n is the dimension of the vector output. A scalar function will always output a scalar result, so it's range is just R. 

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

A vector-valued function such as:

r(t) = f(t) i + g(t) j 

can be graphed in GeoGebra by writing:

(f(t), g(t)).

Final Vector Valued Function Quiz

Question

What is a vector-valued function?

Show answer

Answer

A vector-valued function is a function that takes a scalar value as input and gives a vector as output.

Show question

Question

What does a vector-valued function look like in column vector form?

Show answer

Answer

\[ \vec{r}(t) = \begin{bmatrix} f(t) \\ g(t) \end{bmatrix}. \]

Show question

Question

What does a vector-valued function look like in component form?

Show answer

Answer

\[ \vec{r}(t) = f(t) \vec{i} + g(t) \vec{j}. \] 

Show question

Question

In the equation: 
\[ \vec{r}(t) = \begin{bmatrix} f(t) \\ g(t) \end{bmatrix} = f(t) \vec{i} + g(t) \vec{j}. \] 
What type of equations is \(f(t)\) and \(g(t)\)?

Show answer

Answer

Parametric equations.

Show question

Question

What is the domain of a 2 dimensional vector valued function?

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Answer

 \(\mathbb{R}.\) 

Show question

Question

What is the range of a 2 dimensional vector valued function?

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Answer

\(\mathbb{R}^n.\)

Show question

Question

What is the vector-valued function for a line going through a point \(a\) parallel to the vector \(\vec{b}?\)

Show answer

Answer

\[ \vec{r}(t) = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} + t \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}. \] 

Show question

Question

What is the vector-valued function for a circle with radius \(a\)?

Show answer

Answer

\[ \vec{r}(t) = \begin{bmatrix} a \cos{t} \\ a \sin{t} \end{bmatrix}. \] 

Show question

Question

What is the vector-valued function for an ellipse with \(x\)-intercept \(a\) and \(y\)-intercept \(b\)?

Show answer

Answer

\[ \vec{r}(t) = \begin{bmatrix} a \cos{t} \\ b \sin{t} \end{bmatrix}. \] 

Show question

Question

What is the vector equation for a spiral?

Show answer

Answer

\[ \vec{r}(t) = \begin{bmatrix} a t \cos{t} \\ b t \sin{t} \end{bmatrix}. \]

Show question

Question

How do you sketch a graph of a vector-valued function?

Show answer

Answer

Create a table of values with columns \(t, x\) and \(y,\) and sketch these points. Then draw the curve between them.

Show question

Question

What is the formula for the arc length between points \(a\) and \(b\) of a vector-valued function?

Show answer

Answer

\[ L = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2} \, \mathrm{d}t. \] 

Show question

Question

What is the formula for the derivative of a vector-valued function?

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Answer

\[ \frac{\mathrm{d}\vec{r}}{\mathrm{d}t}(t) = \frac{\mathrm{d}f}{\mathrm{d}t}(t) \vec{i} + \frac{\mathrm{d}g}{\mathrm{d}\mathrm{t}}(t) \vec{j}. \]

Show question

Question

What is the derivative of the position vector \(\vec{s}(t)?\)

Show answer

Answer

The velocity vector \(\vec{v}(t).\)

Show question

Question

What is the derivative of the velocity vector \(\vec{v}(t)?\) 

Show answer

Answer

The acceleration vector \(\vec{a}(t).\)

Show question

Question

What is the definition of the limit of \(\vec{r}(t)\) as \(t\) approaches \(c?\)

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Answer

A vector \( \lim\limits_{t \rightarrow c} \vec{r}(t) = \vec{L} \) such that for all \( \epsilon > 0, \) there exists \( \delta > 0 \) such that for any \( t \neq c, \) if \( | t - c | < \delta ,\) then \( | \vec{r}(t) - \vec{L} | < \epsilon. \) 

Show question

Question

What is the sum rule of limits in a vector-valued function?

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Answer

\[ \lim\limits_{t \rightarrow c} (\vec{r}_1(t) + \vec{r}_2(t)) = \lim\limits_{t \rightarrow c} \vec{r}_1(t) + \lim\limits_{t \rightarrow c} \vec{r}_2(t) \]

Show question

Question

What is the scalar multiple rule of limits in a vector-valued function?

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Answer

\( \lim\limits_{t \rightarrow c} (a \vec{r}_1(t)) = a \lim\limits_{t \rightarrow c} (\vec{r}_1(t)). \)

Show question

Question

What is the dot product rule of limits in a vector-valued function?

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Answer

\[ \lim\limits_{t \rightarrow c} (\vec{r}_1(t) \cdot \vec{r}_2(t)) = \left(\lim\limits_{t \rightarrow c} \vec{r}_1(t)\right) \cdot \left( \lim\limits_{t \rightarrow c} \vec{r}_2(t) \right) \]

Show question

Question

What is the scalar product rule of limits in a vector-valued function?

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Answer

\[ \lim\limits_{t \rightarrow c} (\vec{r}_1(t) \times \vec{r}_2(t)) = \left(\lim\limits_{t \rightarrow c} \vec{r}_1(t)\right) \times \left( \lim\limits_{t \rightarrow c} \vec{r}_2(t) \right). \]

Show question

Question

What does it mean for \(\vec{r}(t)\) to be continuous at \(c?\)

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Answer

\(\lim\limits_{t \rightarrow c} \vec{r}(t) = \vec{r}(c).\)

Show question

Question

What does it mean if \(\vec{r}(t)\) is continuous on \(I\)?

Show answer

Answer

At every point \(t' \in I,\) \( \lim\limits_{t \rightarrow t'} \vec{r}(t) = \vec{r}(t').\)

Show question

Question

Is the vector-valued function \( \vec{r}(t) = f(t) \vec{i} + g(t) \vec{j}  \) continuous on the real line, if

\[ \begin{align} f(x) & = \frac{t^3 - 3x}{t^2 - 9} \\ g(t) & = t^2 + 6t + 2 ? \end{align} \]  

Show answer

Answer

Yes.

Show question

Question

What criteria determines whether the limit as \(t\) goes to \(c\) of a vector-valued function \( \vec{r}(t) = f(t) \vec{i} + g(t) \vec{j}\) exists?

Show answer

Answer

The limits \[\lim\limits_{t \rightarrow c} f(t), \quad \lim\limits_{t \rightarrow c} g(t),\text{ and } \lim\limits_{t \rightarrow c} h(t) \] must exist.

Show question

Question

If  \( \vec{r}(t) = f(t) \vec{i} + g(t) \vec{j} \) and \( \lim\limits_{t \rightarrow c} \vec{r}(t) \) exists, what will the limit be?

Show answer

Answer

\[ \lim\limits_{t \rightarrow c} \vec{r}(t) = \lim\limits_{t \rightarrow c} f(t) \vec{i} + \lim\limits_{t \rightarrow c} g(t) \vec{j}.\]

Show question

Question

Does the limit as \(t\) goes to \(0\) exist in the following function: \[ \vec{r}(t) = \begin{bmatrix} \frac{\sin{t}}{t} \\ 4 \\ \frac{t^2 - 4t}{t} \end{bmatrix}? \] 

Show answer

Answer

Yes.

Show question

Question

Does the limit as \(t\) goes to \(0\) exist in the following vector-valued function:

\[ \vec{r}(t) = 7 \vec{i} + \sin{\frac{1}{t}} \vec{j}. \]

Show answer

Answer

No.

Show question

Question

When is a vector-valued function continuous?

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Answer

When the component functions are continuous.

Show question

Question

In the following formula, what is L?

for all \( \epsilon > 0, \) there exists \( \delta > 0 \) such that for any \( t \neq c, \) if \( | t - c | < \delta ,\) then \( | \vec{r}(t) - \vec{L} | < \epsilon. \)

Show answer

Answer

The limit of \(\vec{r}(t)\) as \(t\) goes to \(c.\)

Show question

Question

What word can be used to describe \(t\) at \(c\) if the following formula is true?

\(\lim\limits_{t \rightarrow c} \vec{r}(t) = \vec{r}(c).\)

Show answer

Answer

\(\vec{r}(t)\) is continuous at \(c.\)

Show question

Question

The definition of the derivative of a vector-valued function is:

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Answer

The derivative of a vector-valued function \( \vec{r}(t) \) is the limit of the difference quotient:

\[ \begin{align}
\vec{r}~'(t) &= \frac{\mathrm{d}}{\mathrm{d}t} \vec{r}(t) \\
&= \lim_{\Delta t \to 0} \frac{(t + \Delta t) - \vec{r}(t)}{\Delta t},
\end{align} \]

if the limit exists.

  • If \( \vec{r}~'(t) \) does exist, then \( \vec{r}(t) \) is differentiable at \(t\).
  • If \( \vec{r}~'(t) \) exists for all \(t\) in the open interval of \( (a, b) \), then \( \vec{r}(t) \) is differentiable over the open interval of \( (a, b) \).

However, for \( \vec{r}(t) \) to be differentiable over the closed interval of \( [a, b] \), then the following two limits must also exist:

\[ \begin{align}
\vec{r}~'(a) &= \lim_{\Delta t \to 0^{+}} \frac{\vec{r}(a + \Delta t) - \vec{r}(a)}{\Delta t} &(1) \\
\\
\vec{r}~'(b) &= \lim_{\Delta t \to 0^{-}} \frac{\vec{r}(b + \Delta t) - \vec{r}(b)}{\Delta t} &(2)
\end{align} \]

Show question

Question

State the theorem: Vector-Valued Derivatives are Computed Component Wise.

Show answer

Answer

  • A vector-valued function:

    \[ \vec{r}(t) =
    \begin{bmatrix}
    x(t) \\
    y(t)
    \end{bmatrix} \]
    is differentiable if and only if each of its components is differentiable.

  • This is represented mathematically as:

    \[ \vec{r}~'(t) = \frac{\mathrm{d}}{\mathrm{d}t} \vec{r}(t) =
    \begin{bmatrix}
    x'(t) \\
    y'(t)
    \end{bmatrix} \]

Show question

Question

When a vector-valued function represents the position of an object at a given point in time, the derivative represents the object's _ at that same point in time.

Show answer

Answer

velocity

Show question

Question

The differentiation rules you learned for scalar functions extend to vector-valued functions. Specifically, _, _, _, and _ all extend to vector-valued functions.

Show answer

Answer

The constant multiple rule

Show question

Question

In the case of the product rule, there are three extensions to vector-valued functions.

What are they?

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Answer

  1. the product rule – for a scalar function multiplied by a vector-valued function,

  2. the dot product rule – for the dot product of two vector-valued functions, and

  3. the cross product rule – for the cross product of two vector-valued functions.

Show question

Question

What is the Constant Multiple Rule of Vector-Valued Functions?

Show answer

Answer

For any constant \(c\) and a vector-valued function \( \vec{r}(t) \):

\[ \left( c \vec{r}(t) \right)' = c \vec{r}~'(t) \]

Show question

Question

What is the Sum and Difference Rule of Vector-Valued Functions?

Show answer

Answer

For two vector-valued functions, \( \vec{r_1}(t) \) and \( \vec{r_2}(t) \):

\[ \left( \vec{r_1}(t) \pm \vec{r_2}(t) \right)' = \vec{r_1}~'(t) \pm \vec{r_2}'(t) \]

Show question

Question

What is the Product Rule of a Scalar Function and a Vector-Valued Function?

Show answer

Answer

For any differentiable scalar function \( f(t) \) multiplied by a vector-valued function \( \vec{r}(t) \):

\[ \left( f(t) \vec{r}(t) \right)' = f(t) \vec{r}~'(t) + f'(t) \vec{r}(t) \]

Show question

Question

What is the Dot Product Rule of Vector-Valued Functions?

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Answer

For two differentiable vector-valued functions, \( \vec{r_1}(t) \) and \( \vec{r_2}(t) \):

\[ \left( \vec{r_1}(t) \cdot \vec{r_2}(t) \right)' = \vec{r_1}'(t) \cdot \vec{r_2}(t) + \vec{r_1}(t) \cdot \vec{r_2}'(t) \]

Show question

Question

What is the Cross Product Rule of Vector-Valued Functions?

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Answer

For two differentiable vector-valued functions, \( \vec{r_1}(t) \) and \( \vec{r_2}(t) \):

\[ \left( \vec{r_1}(t) \times \vec{r_2}(t) \right)' = \vec{r_1}'(t) \times \vec{r_2}(t) + \vec{r_1}(t) \times \vec{r_2}'(t) \]

Show question

Question

What is the Chain Rule of Vector-Valued Functions?

Show answer

Answer

For any differentiable scalar function \( f(t) \):

\[ \left( \vec{r}(f(t)) \right)' = \vec{r}~'(f(t)) \cdot f'(t) \]

Show question

Question

The derivative of a vector-valued function, \( \vec{r}~'(t_0) \), however, exhibits an important geometric property:

Show answer

Answer

  • It provides a tangent vector to the curve and

  • It points in the direction tangent to the path traced by \( \vec{r}(t) \) at \(t = t_0 \).

Show question

Question

What is the tangent vector to a curve?

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Answer

The tangent vector is a directional vector for the tangent line to the curve. Its parametrization is:

\[ \text{Tangent line at } \vec{r}(t_{0}): \vec{L} = \vec{r}(t_{0}) + t \vec{r}~'(t_{0}) \]

Show question

Question

What is a unit tangent vector to a curve?

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Answer

The unit tangent vector is precisely what it sounds like: a unit vector that is tangent to the curve. It is formally defined as:


  1. Let \(C\) be some curve defined by a vector-valued function \( \vec{r}(t) \).
  2. Assume that \( \vec{r}~'(t) \) exists when \( t = t_{0} \).
  3. A tangent vector \( \vec{v}(t) \) at \( t = t_{0} \) is any vector such that – when the tail of the vector is placed at point \( \vec{r}(t_{0}) \) on the graph – vector \( \vec{v}(t) \) is tangent to curve \(C\).
  4. Vector \( \vec{r}~'(t_{0}) \) is an example of a tangent vector at point \( t = t_{0} \).
  5. Finally, assume that \( \vec{r}~'(t) \neq 0 \).


If all of the above is true, then the unit tangent vector at \(t\) is defined as:

\[ \vec{T}(t) = \frac{\vec{r}~'(t)}{\left|\vec{r}~'(t)\right|} \]

if \( \left|\vec{r}~'(t)\right| \neq 0 \).

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Question

The steps to calculate a unit tangent vector are:

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Answer

  1. find the derivative of \( \vec{r}~'(t) \),

  2. calculate the magnitude of the derivative, and

  3. divide the derivative by its magnitude.

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Question

What is the definition of the integral of a vector-valued function?

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Answer

If two functions, \( f(t) \) and \( g(t) \), are integrable scalar functions over the closed interval of \( [a, b] \), then


  • The indefinite integral of a vector-valued function: \( \vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} \) is:
    \[ \int[f(t) \vec{i} + g(t) \vec{j}] \mathrm{d}t = \left[ \int f(t) \mathrm{d}t \right] \vec{i} + \left[ \int g(t) \mathrm{d}t \right] \vec{j} \]
  • The definite integral of a vector-valued function: \( \vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} \) is:
    \[ \int_{a}^{b}[f(t) \vec{i} + g(t) \vec{j}] \mathrm{d}t = \left[ \int_{a}^{b} f(t) \mathrm{d}t \right] \vec{i} + \left[ \int_{a}^{b} g(t) \mathrm{d}t \right] \vec{j} \]

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Question

What is a constant vector?

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Answer

A constant vector is a vector that does not depend on \(t\).

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Question

What is the fundamental theorem of calculus for vector-valued functions?

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Answer

If the vector-valued function \( \vec{r}(t) \) is continuous over the closed interval of \( [a, b] \), and \( \vec{R}(t) \) is an antiderivative of \( \vec{r}(t) \), then:

\[ \int_{a}^{b} \vec{r}(t) \mathrm{d}t = \vec{R}(b) - \vec{R}(a) \]

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Question

The output of a vector-valued function is a ____.

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Answer

vector.

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Question

Consider the vector-valued function
\[ \vec{r} (t) = \begin{bmatrix} \sin{t} \\ t^2-t \end{bmatrix}.\]
This vector-valued function is written in ____ form.

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Answer

column vector.

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