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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Vector Addition

Find the sum of the vectors given below

$\vec{A}= 4 \hat{x}-\hat{y}, \vec{B}=-3\hat{x}+10\hat{y}$

Possible Answers:

$\vec{A}+\vec{B}= -\hat{x}-9\hat{y}$

$\vec{A}+\vec{B}= 10$

$\vec{A}+\vec{B}= \hat{x}+9\hat{y}$

$\vec{A}+\vec{B}= 3\hat{x}+7\hat{y}$

Correct answer:

$\vec{A}+\vec{B}= \hat{x}+9\hat{y}$

Explanation:

When adding vectors, it is important to note that the summation only occurs between terms that have the same coordinate direction $\left( \hat{x}, \hat{y}, etc. \right)$ . Therefore, we find

$\vec{A}+\vec{B}= \left(4\hat{x}-1\hat{y} \right ) + \left( -3\hat{x}+10\hat{y}\right) = \hat{x}(4+(-3))+\hat{y}(-1+10) =\hat{x}+9\hat{y}$

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Example Question #5 : Vector Addition

Find the sum of the two vectors.

u=<1,0,0>

v=<0,0,3>

Possible Answers:

<1,0,3>

<1,0,-3>

Correct answer:

<1,0,3>

Explanation:

The sum of two vectors <a_1,a_2,a_3> and <b_1,b_2,b_3> is defined as

<a_1+b_1,a_2+b_2,a_3+b_3>

For the vectors

u=<1,0,0>

v=<0,0,3>

u+v=<1,0,0>+<0,0,3>

=<1+0,0+0,0+3>

=<1,0,3>

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Example Question #1 : Vector Addition

Given vectors

v=(1,-1,2)

w=(0,5,3)

find v+w .

Possible Answers:

v+w=(1,4,5)

v+w=(0,5,3)

v+w=(5,1,4)

v+w=(1,2,3)

Correct answer:

v+w=(1,4,5)

Explanation:

To find the sum v+w , we add their components:

v+w=(1,-1,2)+(0,5,3)=(1+0,-1+5,2+3)=(1,4,5)

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Example Question #1 : Vector Addition

Given the vectors

v=(1,3)

w=(5,2)

find the sum v+w .

Possible Answers:

v+w=(6,5)

v+w=(-4,1)

v+w=(4,-1)

v+w=(5,6)

Correct answer:

v+w=(6,5)

Explanation:

To find the sum v+w of the vectors

v=(1,3)

w=(5,2)

we add their components:

v+w=(1,3)+(5,2)=(1+5,3+2)=(6,5)

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Example Question #8 : Vector Addition

Add the following vectors:

$\mathbf{u}+\mathbf{v}$

Where

$\mathbf{u}=12\mathbf{i}-7\mathbf{j}+10\mathbf{k}$ , $\mathbf{v}=3\mathbf{i}+8\mathbf{j}-4\mathbf{k}$

Possible Answers:

$8\mathbf{i}+\mathbf{j}+13\mathbf{k}$

$9\mathbf{i}-15\mathbf{j}+14\mathbf{k}$

$15\mathbf{i}+\mathbf{j}+6\mathbf{k}$

$7\mathbf{i}+16\mathbf{j}-3\mathbf{k}$

Correct answer:

$15\mathbf{i}+\mathbf{j}+6\mathbf{k}$

Explanation:

Vector addition is done as follows:

$\mathbf{u}+\mathbf{v}=(u_x+v_x)\mathbf{i}+(u_y+v_y)\mathbf{j}+(u_z+v_z)\mathbf{k}$

For this problem:

$\mathbf{u}+\mathbf{v}=(12+3)\mathbf{i}+((-7)+8)\mathbf{j}+(10+(-4))\mathbf{k}= 15\mathbf{i}+\mathbf{j}+6\mathbf{k}$

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Example Question #1 : Vector Addition

Add the vectors 3x+9y given: $x=\left \langle 3,-2\right \rangle$ and $y=\left \langle -3,-6\right \rangle$

Possible Answers:

$\left \langle 0,-72\right \rangle$

$\left \langle -6,-60\right \rangle$

$\left \langle18,-60 \right \rangle$

$\left \langle-18,-60 \right \rangle$

$\left \langle -6,-72\right \rangle$

Correct answer:

$\left \langle-18,-60 \right \rangle$

Explanation:

Multiply the vectors with the constants first.

Evaluate .

$3\left \langle 3,-2\right \rangle = \left \langle 9,-6\right \rangle$

Evaluate .

$9\left \langle -3,-6\right \rangle =\left \langle -27, -54 \right \rangle$

Add the vectors 3x+9y .

$\left \langle 9,-6\right \rangle+\left \langle -27, -54 \right \rangle = \left \langle-18,-60 \right \rangle$

The answer is: $\left \langle-18,-60 \right \rangle$

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Example Question #10 : Vector Addition

Given the vectors

$\small v=(\pi,e)$

$\small w=(1,2)$

find the sum $\small v+w$ .

Possible Answers:

$\small \small v+w=(\pi+1,e+3)$

$\small \small v+w=(\pi+1,e^2)$

$\small \small v+w=(e+2,\pi+1)$

$\small v+w=(\pi+1,e+2)$

Correct answer:

$\small v+w=(\pi+1,e+2)$

Explanation:

Given the vectors

$\small v=(\pi,e)$

$\small w=(1,2)$

we can find the sum $\small v+w$ by adding component by component:

$\small \small v+w=(\pi,e)+(1,2)=(\pi+1,e+2)$

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Example Question #2211 : Calculus 3

Given the vectors

$\small \small \small v=\left(\frac{1}{2},\frac{3}{4}\right)$

$\small \small w=\left(\frac{5}{2},\frac{5}{4}\right)$

find the sum $\small v+w$ .

Possible Answers:

$\small \small \small v+w=(2,3)$

$\small \small v+w=(3,2)$

$\small \small \small v+w=(-2,2)$

$\small \small \small v+w=(0,2)$

Correct answer:

$\small \small v+w=(3,2)$

Explanation:

Given the vectors

$\small \small \small v=\left(\frac{1}{2},\frac{3}{4}\right)$

$\small \small w=\left(\frac{5}{2},\frac{5}{4}\right)$

we can find the sum $\small v+w$ by adding component by component:

$\small \small \small v+w=(1/2,3/4)+(5/2,5/4)=(1/2+5/2,3/4+5/4)$

=(6/2,8/4)=(3,2)

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Example Question #11 : Vector Addition

Given the vectors

$\small \small \small \small v=\left(\frac{1}{2},2\right)$

$\small \small \small \small w=\left(\frac{1}{4},-5\right)$

find the sum $\small v+w$ .

Possible Answers:

$\small \small \small v+w=(3/4,3)$

$\small \small v+w=(3/4,-3)$

$\small \small \small v+w=(5/8,-3)$

$\small \small \small v+w=(3/4,2)$

Correct answer:

$\small \small v+w=(3/4,-3)$

Explanation:

Given the vectors

$\small \small \small \small v=\left(\frac{1}{2},2\right)$

$\small \small \small \small w=\left(\frac{1}{4},-5\right)$

we can find the sum $\small v+w$ by adding component by component:

$\small \small \small \small \small v+w=(1/2,2)+(1/4,-5)=(1/2+1/4,2-5)$

$\small =(3/4,-3)$

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Example Question #211 : Vectors And Vector Operations

Given the vectors

$\small \small \small \small \small \small v=(-1,2,\pi^2)$

$\small \small \small \small \small \small \small w=(1,-2,\pi)$

find the sum $\small v+w$ .

Possible Answers:

$\small \small \small \small \small \small v+w=(0,0,\pi(\pi+1))$

$\small \small \small \small \small \small \small v+w=(-2,-4,\pi(\pi-1))$

$\small \small \small \small \small \small \small v+w=(0,0,\pi(\pi-1))$

$\small \small \small \small \small \small \small \small v+w=(0,0,\pi(1-\pi))$

Correct answer:

$\small \small \small \small \small \small v+w=(0,0,\pi(\pi+1))$

Explanation:

Given the vectors

$\small \small \small \small \small \small v=(-1,2,\pi^2)$

$\small \small \small \small \small \small \small w=(1,-2,\pi)$

we can find the sum $\small v+w$ by adding component by component:

$\small \small \small \small \small \small \small \small v+w=(-1,2,\pi^2)+(1,-2,\pi)=(-1+1,2-2,\pi^2+\pi)$

$\small \small \small \small \small =(0,0,\pi(\pi+1))$

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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Vector Addition

Example Question #5 : Vector Addition

Example Question #1 : Vector Addition

Example Question #1 : Vector Addition

Example Question #8 : Vector Addition

Example Question #1 : Vector Addition

Example Question #10 : Vector Addition

Example Question #2211 : Calculus 3

Example Question #11 : Vector Addition

Example Question #211 : Vectors And Vector Operations

All Calculus 3 Resources

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