Calculus 3 : Vector Addition

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #1 : Vector Addition

Let \displaystyle \vec{a}=(1,2,3)\displaystyle \vec{b}=(10,-10,3), and \displaystyle \vec{c}=(0,0,1),

find \displaystyle \vec{a}+\vec{b}+\vec{c}

Possible Answers:

\displaystyle (7, -8,11)

\displaystyle (11, -8, 7)

\displaystyle (11, 7, -8)

\displaystyle (0,0,1)

\displaystyle (11, 0, 1)

Correct answer:

\displaystyle (11, -8, 7)

Explanation:

In order to find \displaystyle \vec{a}+\vec{b}+\vec{c}, we need to add like components. \displaystyle \vec{a}+\vec{b}+\vec{c}=(1+10+0,2-10+0,3+3+1)=(11,-8,7)

Example Question #1 : Vector Addition

Given the vectors 

\displaystyle \small v=(2,3) and \displaystyle \small w=(2,100), compute \displaystyle \small v+w.

Possible Answers:

\displaystyle \small \small \small v+w=(0,100)

\displaystyle \small \small \small v+w=(104,5)

\displaystyle \small \small v+w=(103,4)

\displaystyle \small \small v+w=(4,103)

Correct answer:

\displaystyle \small \small v+w=(4,103)

Explanation:

To add two vectors \displaystyle \small v,w, we simply add their components:

\displaystyle \small \small \small v+w=(2,3)+(2,100)=(2+2,3+100)=(4,103)

Example Question #2 : Vector Addition

Given the vectors 

\displaystyle \small \small v=(0,10) and \displaystyle \small \small w=(-10,0), compute \displaystyle \small v+w.

Possible Answers:

\displaystyle \small \small \small \small v+w=(10,10)

\displaystyle \small \small \small \small \small v+w=(10,-10)

\displaystyle \small \small \small \small v+w=(-10,10)

\displaystyle \small \small \small \small v+w=(0,0)

Correct answer:

\displaystyle \small \small \small \small v+w=(-10,10)

Explanation:

To add two vectors \displaystyle \small v,w, we simply add their components:

\displaystyle \small \small \small \small v+w=(0,10)+(-10,0)=(0-10,10+0)=(-10,10)

Example Question #2211 : Calculus 3

Find the sum of the vectors given below

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

Possible Answers:

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

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

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

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

Correct answer:

\displaystyle \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 \displaystyle \left( \hat{x}, \hat{y}, etc. \right).  Therefore, we find 

\displaystyle \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}

Example Question #5 : Vector Addition

Find the sum of the two vectors. 

\displaystyle u=< 1,0,0>

\displaystyle v=< 0,0,3>

Possible Answers:

\displaystyle < 1,0,3>

\displaystyle 4

\displaystyle 0

\displaystyle < 1,0,-3>

Correct answer:

\displaystyle < 1,0,3>

Explanation:

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

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

For the vectors 

\displaystyle u=< 1,0,0>

\displaystyle v=< 0,0,3>

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

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

\displaystyle =< 1,0,3>

Example Question #1 : Vector Addition

Given vectors

\displaystyle v=(1,-1,2)

\displaystyle w=(0,5,3)

find \displaystyle v+w.

Possible Answers:

\displaystyle v+w=(5,1,4)

\displaystyle v+w=(1,2,3)

\displaystyle v+w=(1,4,5)

\displaystyle v+w=(0,5,3)

Correct answer:

\displaystyle v+w=(1,4,5)

Explanation:

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

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

Example Question #1 : Vector Addition

Given the vectors

\displaystyle v=(1,3)

\displaystyle w=(5,2)

find the sum \displaystyle v+w.

Possible Answers:

\displaystyle v+w=(6,5)

\displaystyle v+w=(-4,1)

\displaystyle v+w=(4,-1)

\displaystyle v+w=(5,6)

Correct answer:

\displaystyle v+w=(6,5)

Explanation:

To find the sum \displaystyle v+w of the vectors

\displaystyle v=(1,3)

\displaystyle w=(5,2)

 we add their components:

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

Example Question #8 : Vector Addition

Add the following vectors:

\displaystyle \mathbf{u}+\mathbf{v}

Where

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

Vector addition is done as follows:

\displaystyle \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:

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

Example Question #1 : Vector Addition

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Multiply the vectors with the constants first.

Evaluate \displaystyle 3x.

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

Evaluate \displaystyle 9y.

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

Add the vectors \displaystyle 3x+9y.

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

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

Example Question #10 : Vector Addition

Given the vectors 

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

\displaystyle \small w=(1,2)

find the sum \displaystyle \small v+w.

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

Given the vectors 

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

\displaystyle \small w=(1,2)

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

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

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