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9.4 Defining and Differentiating Vector-Valued Functions

6 min readjanuary 24, 2023

So far in this unit, we’ve been delving deep into the functions of parametric equations. Now, we’re taking this knowledge and talking about vector-valued functions!

↖️ Vector Review

A vector is a quantity that has both direction and magnitude. Magnitude refers to the length of a vector. All vectors have horizontal and vertical components. They are written and defined based on these two components. For example, a vector with a horizontal component of 5 and a vertical component of 4 would be written as <5,4><5, 4> . While there are a couple of other methods of writing vectors, this kind of notation is used most frequently on the AP Calculus BC Exam!

Untitled

Image Courtesy of LaTeX Stack Exchange

Every vector is drawn with a head and a tail. The tail of the vector is the point at which the vector originates. If there is no additional information given, we can assume that the vector originates from (0,0) and that the tail of the vector is at (0,0). The head of the vector is the final point of the vector. In diagrams, the head of the vector is represented with an arrowhead.

Untitled

Image Courtesy of Labster

The magnitude of the vector can be calculated based on the horizontal and vertical components of the vector. Let’s find the magnitude of the vector we used earlier, <5,4><5,4>. This vector has a horizontal component of 5 and a vertical component of 4. We can find the magnitude by using the Pythagorean theorem and calculating for the hypotenuse. The Pythagorean theorem is a2+b2=c2a^2 + b^2=c^2. In this scenario, a=5a = 5 and b=4b=4. Therefore, c=41c= \sqrt{41}. The magnitude of our vector is 41\sqrt{41}. The magnitude of the vector is written as ‖𝑣‖.

We can also calculate the direction of the vector from its horizontal and vertical components. We know that tan(θ)=vertical componenthorizontal componenttan(θ)= \frac{\text{vertical component}}{\text{horizontal component}}. For our example vector from earlier, tan(θ)=45tan(θ)= \tfrac{4}{5}. Using inverse tangent, we can calculate that θ=0.675θ= 0.675.

Untitled

Image Courtesy of Study.com

We have now seen that we can calculate the magnitude and direction of a vector given its horizontal and vertical components. Similarly, we can also calculate the horizontal and vertical components from the magnitude and direction of the vector.


📐 Vector-Valued Functions

Think of r(t)r(t) as a vector composed of the two parametric functions f(t)f(t) and g(t)g(t). We can write the vector as r(t)=<f(t),g(t)>.r(t)=<f(t), g(t)>. If we wanted to differentiate this vector-valued function, we can write the derivative of the function as r(t)=<f(t),g(t)>r'(t) = <f'(t), g'(t)>. Normal derivation rules apply to vector-valued functions! These rules are summarized in the image below.

maxresdefault.jpg

Image Courtesy of the Algebros

✏️ Vector-Valued Functions Walkthrough

Let’s do a quick example!

r(t)=<4t2,7t7>r(t) = <4t^2, 7t^7>

Find r(4).r'(4).

Since we are asked to find r(4)r'(4), let’s begin by finding r(t)r'(t). We do this by taking the derivative of each of the components. We can start with the horizontal component, 4t24t^2. The derivative of 4t24t^2 with respect to tt is 8t8t.

The vertical component of r(t)r(t) is 7t77t^7. The derivative of 7t77t^7 with respect to tt is 49t649t^6.

Now, let’s put it all together!

r(t)=<8t,49t6>r'(t) = <8t, 49t^6>

We can then substitute tt for 44 to find r(4)r’(4).

r(4)=<8(4),49(4)6>r'(4) = <8(4), 49(4)^6>

And now we simplify!

r(2)=<32,200704>r'(2) = < 32, 200704>

Now that we know a bit about vectors and vector-valued functions, let’s amp it up and try some practice problems :)


🤔 Practice Problems with Vector-Valued Functions

We have to two practice problems embedded in this study guide for you!

💭 Vector Practice Problem 1

Find r(2)r'(2) given that

r(t)=<4sin(5t),7cos(2t2)>r(t) = <4sin(5t), 7cos(2t^2)>

Let’s begin by finding the derivative of each of the components. We can start with the horizontal component, 4sin(5t)4sin(5t). The derivative of 4sin(5t)4sin(5t) with respect to tt is 20cos(5t)20cos(5t).

The vertical component of r(t)r(t) is 7cos(2t2)7cos(2t^2). The derivative of 7cos(2t2)7cos(2t^2) with respect to tt is 28tsin(2t2)-28tsin(2t^2).

Now, let’s put it all together!

r(t)=<20cos(5t),28tsin(2t2)>r'(t) = <20cos(5t), -28tsin(2t^2)>

We can then substitute tt for 22.

r(2)=<20cos(10),56sin(8)>r'(2) = <20cos(10), -56sin(8)>

All that’s left to do is simplify!

r(2)=<16.781,255.404>r'(2) = < -16.781, -255.404>

Great work! Let’s try another practice problem.

💭 Vector Practice Problem 2

A particle moves along a path so that its position at any given time where t≥0 is given by (4sin(t),4cos(t)).(4sin(t), 4 cos(t)). Find the particle’s velocity vector at t=π4t=\tfrac{π}{4}.

Here’s the solution!

A particle’s velocity vector is the first derivative of its position vector.

Therefore, v(t)=<dxdt,dydt>v(t)=<\tfrac{dx}{dt}, \tfrac{dy}{dt}>. To find the particle’s velocity vector at t=π4t=\tfrac{π}{4}, we can begin by taking the first derivative of the position vector.

We can start with the horizontal component, 4sin(t)4sin(t). The derivative of 4sin(t)4sin(t) with respect to tt is 4cos(t)4cos(t).

The vertical component of r(t)r(t) is 4cos(t)4cos(t). The derivative of 4cos(t)4cos(t) with respect to tt is 4sin(t)-4sin(t).

Like earlier, let’s put it all together!

r(t)=<4cos(t),4sin(t)>r'(t) = <4cos(t), -4sin(t)>

We can then substitute tt for π4\tfrac{π}{4}.

r(π4)=<4cos(π4),4sin(π4)>r'(\tfrac{π}{4}) = <4cos(\tfrac{π}{4}), -4sin(\tfrac{π}{4})>

All that’s left to do is simplify!

r(π4)=<22,22>r'(\tfrac{π}{4}) = < 2\sqrt{2}, -2\sqrt{2}>

As you keep practicing, solving these problems will get easier and easier! Like always, practice is key. 🔑

You got this!!

Key Terms to Review (10)

a(t)

: The term "a(t)" represents the acceleration of an object at a specific time. It measures how quickly the velocity of an object is changing.

Acceleration Vector

: The acceleration vector represents the rate at which an object's velocity is changing over time. It includes both the magnitude (how fast the object is accelerating) and direction of acceleration.

dy/dx

: The derivative of a function y with respect to x, representing the rate of change of y with respect to x.

Motion

: Motion refers to the change in position of an object over time. It involves concepts such as distance, displacement, speed, velocity, and acceleration.

Parametric Functions

: Parametric functions are a way to represent curves or graphs using two separate equations, one for the x-coordinate and one for the y-coordinate. These equations are typically in terms of a third variable called the parameter.

r(t)

: The position vector that describes an object's location in space as a function of time.

Tangent Line

: A tangent line is a straight line that touches a curve at only one point without crossing through it. In calculus, we use tangent lines to approximate curves and find instantaneous rates of change.

v(t)

: The instantaneous velocity or speed of an object at any given time t.

Velocity

: Velocity is the rate at which an object's position changes with respect to time. It includes both speed and direction.

Velocity Vector

: A velocity vector represents both the speed and direction of an object's motion. It is a mathematical representation that combines magnitude (speed) and direction into a single quantity.

9.4 Defining and Differentiating Vector-Valued Functions

6 min readjanuary 24, 2023

So far in this unit, we’ve been delving deep into the functions of parametric equations. Now, we’re taking this knowledge and talking about vector-valued functions!

↖️ Vector Review

A vector is a quantity that has both direction and magnitude. Magnitude refers to the length of a vector. All vectors have horizontal and vertical components. They are written and defined based on these two components. For example, a vector with a horizontal component of 5 and a vertical component of 4 would be written as <5,4><5, 4> . While there are a couple of other methods of writing vectors, this kind of notation is used most frequently on the AP Calculus BC Exam!

Untitled

Image Courtesy of LaTeX Stack Exchange

Every vector is drawn with a head and a tail. The tail of the vector is the point at which the vector originates. If there is no additional information given, we can assume that the vector originates from (0,0) and that the tail of the vector is at (0,0). The head of the vector is the final point of the vector. In diagrams, the head of the vector is represented with an arrowhead.

Untitled

Image Courtesy of Labster

The magnitude of the vector can be calculated based on the horizontal and vertical components of the vector. Let’s find the magnitude of the vector we used earlier, <5,4><5,4>. This vector has a horizontal component of 5 and a vertical component of 4. We can find the magnitude by using the Pythagorean theorem and calculating for the hypotenuse. The Pythagorean theorem is a2+b2=c2a^2 + b^2=c^2. In this scenario, a=5a = 5 and b=4b=4. Therefore, c=41c= \sqrt{41}. The magnitude of our vector is 41\sqrt{41}. The magnitude of the vector is written as ‖𝑣‖.

We can also calculate the direction of the vector from its horizontal and vertical components. We know that tan(θ)=vertical componenthorizontal componenttan(θ)= \frac{\text{vertical component}}{\text{horizontal component}}. For our example vector from earlier, tan(θ)=45tan(θ)= \tfrac{4}{5}. Using inverse tangent, we can calculate that θ=0.675θ= 0.675.

Untitled

Image Courtesy of Study.com

We have now seen that we can calculate the magnitude and direction of a vector given its horizontal and vertical components. Similarly, we can also calculate the horizontal and vertical components from the magnitude and direction of the vector.


📐 Vector-Valued Functions

Think of r(t)r(t) as a vector composed of the two parametric functions f(t)f(t) and g(t)g(t). We can write the vector as r(t)=<f(t),g(t)>.r(t)=<f(t), g(t)>. If we wanted to differentiate this vector-valued function, we can write the derivative of the function as r(t)=<f(t),g(t)>r'(t) = <f'(t), g'(t)>. Normal derivation rules apply to vector-valued functions! These rules are summarized in the image below.

maxresdefault.jpg

Image Courtesy of the Algebros

✏️ Vector-Valued Functions Walkthrough

Let’s do a quick example!

r(t)=<4t2,7t7>r(t) = <4t^2, 7t^7>

Find r(4).r'(4).

Since we are asked to find r(4)r'(4), let’s begin by finding r(t)r'(t). We do this by taking the derivative of each of the components. We can start with the horizontal component, 4t24t^2. The derivative of 4t24t^2 with respect to tt is 8t8t.

The vertical component of r(t)r(t) is 7t77t^7. The derivative of 7t77t^7 with respect to tt is 49t649t^6.

Now, let’s put it all together!

r(t)=<8t,49t6>r'(t) = <8t, 49t^6>

We can then substitute tt for 44 to find r(4)r’(4).

r(4)=<8(4),49(4)6>r'(4) = <8(4), 49(4)^6>

And now we simplify!

r(2)=<32,200704>r'(2) = < 32, 200704>

Now that we know a bit about vectors and vector-valued functions, let’s amp it up and try some practice problems :)


🤔 Practice Problems with Vector-Valued Functions

We have to two practice problems embedded in this study guide for you!

💭 Vector Practice Problem 1

Find r(2)r'(2) given that

r(t)=<4sin(5t),7cos(2t2)>r(t) = <4sin(5t), 7cos(2t^2)>

Let’s begin by finding the derivative of each of the components. We can start with the horizontal component, 4sin(5t)4sin(5t). The derivative of 4sin(5t)4sin(5t) with respect to tt is 20cos(5t)20cos(5t).

The vertical component of r(t)r(t) is 7cos(2t2)7cos(2t^2). The derivative of 7cos(2t2)7cos(2t^2) with respect to tt is 28tsin(2t2)-28tsin(2t^2).

Now, let’s put it all together!

r(t)=<20cos(5t),28tsin(2t2)>r'(t) = <20cos(5t), -28tsin(2t^2)>

We can then substitute tt for 22.

r(2)=<20cos(10),56sin(8)>r'(2) = <20cos(10), -56sin(8)>

All that’s left to do is simplify!

r(2)=<16.781,255.404>r'(2) = < -16.781, -255.404>

Great work! Let’s try another practice problem.

💭 Vector Practice Problem 2

A particle moves along a path so that its position at any given time where t≥0 is given by (4sin(t),4cos(t)).(4sin(t), 4 cos(t)). Find the particle’s velocity vector at t=π4t=\tfrac{π}{4}.

Here’s the solution!

A particle’s velocity vector is the first derivative of its position vector.

Therefore, v(t)=<dxdt,dydt>v(t)=<\tfrac{dx}{dt}, \tfrac{dy}{dt}>. To find the particle’s velocity vector at t=π4t=\tfrac{π}{4}, we can begin by taking the first derivative of the position vector.

We can start with the horizontal component, 4sin(t)4sin(t). The derivative of 4sin(t)4sin(t) with respect to tt is 4cos(t)4cos(t).

The vertical component of r(t)r(t) is 4cos(t)4cos(t). The derivative of 4cos(t)4cos(t) with respect to tt is 4sin(t)-4sin(t).

Like earlier, let’s put it all together!

r(t)=<4cos(t),4sin(t)>r'(t) = <4cos(t), -4sin(t)>

We can then substitute tt for π4\tfrac{π}{4}.

r(π4)=<4cos(π4),4sin(π4)>r'(\tfrac{π}{4}) = <4cos(\tfrac{π}{4}), -4sin(\tfrac{π}{4})>

All that’s left to do is simplify!

r(π4)=<22,22>r'(\tfrac{π}{4}) = < 2\sqrt{2}, -2\sqrt{2}>

As you keep practicing, solving these problems will get easier and easier! Like always, practice is key. 🔑

You got this!!

Key Terms to Review (10)

a(t)

: The term "a(t)" represents the acceleration of an object at a specific time. It measures how quickly the velocity of an object is changing.

Acceleration Vector

: The acceleration vector represents the rate at which an object's velocity is changing over time. It includes both the magnitude (how fast the object is accelerating) and direction of acceleration.

dy/dx

: The derivative of a function y with respect to x, representing the rate of change of y with respect to x.

Motion

: Motion refers to the change in position of an object over time. It involves concepts such as distance, displacement, speed, velocity, and acceleration.

Parametric Functions

: Parametric functions are a way to represent curves or graphs using two separate equations, one for the x-coordinate and one for the y-coordinate. These equations are typically in terms of a third variable called the parameter.

r(t)

: The position vector that describes an object's location in space as a function of time.

Tangent Line

: A tangent line is a straight line that touches a curve at only one point without crossing through it. In calculus, we use tangent lines to approximate curves and find instantaneous rates of change.

v(t)

: The instantaneous velocity or speed of an object at any given time t.

Velocity

: Velocity is the rate at which an object's position changes with respect to time. It includes both speed and direction.

Velocity Vector

: A velocity vector represents both the speed and direction of an object's motion. It is a mathematical representation that combines magnitude (speed) and direction into a single quantity.


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© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.