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Calculus 3 : Vector Addition

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Example Questions

Example Question #1 : Vector Addition

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

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

Possible Answers:

(0,0,1) $\displaystyle (0,0,1)$

$\displaystyle (11, -8, 7)$

$\displaystyle (7, -8,11)$

$\displaystyle (11, 0, 1)$

$\displaystyle (11, 7, -8)$

Correct answer:

$\displaystyle (11, -8, 7)$

Explanation:

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

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Example Question #1 : Vector Addition

Given the vectors

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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

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Example Question #2 : Vector Addition

Given the vectors

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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

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Example Question #1 : Vector Addition

Find the sum of the vectors given below

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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

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Example Question #5 : Vector Addition

Find the sum of the two vectors.

u=<1,0,0> $\displaystyle u=< 1,0,0>$

v=<0,0,3> $\displaystyle v=< 0,0,3>$

Possible Answers:

<1,0,3> $\displaystyle < 1,0,3>$

$\displaystyle 4$

$\displaystyle 0$

<1,0,-3> $\displaystyle < 1,0,-3>$

Correct answer:

<1,0,3> $\displaystyle < 1,0,3>$

Explanation:

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

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

For the vectors

u=<1,0,0> $\displaystyle u=< 1,0,0>$

v=<0,0,3> $\displaystyle v=< 0,0,3>$

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

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

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

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Example Question #1 : Vector Addition

Given vectors

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

$\displaystyle w=(0,5,3)$

find v+w $\displaystyle v+w$.

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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

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Example Question #1 : Vector Addition

Given the vectors

v=(1,3) $\displaystyle v=(1,3)$

w=(5,2) $\displaystyle w=(5,2)$

find the sum v+w $\displaystyle v+w$.

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 v+w $\displaystyle v+w$ of the vectors

v=(1,3) $\displaystyle v=(1,3)$

w=(5,2) $\displaystyle w=(5,2)$

we add their components:

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

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Example Question #8 : Vector Addition

Add the following vectors:

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

Where

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

Vector addition is done as follows:

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

For this problem:

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

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Example Question #1 : Vector Addition

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Multiply the vectors with the constants first.

Evaluate $\displaystyle 3x$.

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

Evaluate $\displaystyle 9y$.

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

Add the vectors 3x+9y $\displaystyle 3x+9y$.

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

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

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Example Question #10 : Vector Addition

Given the vectors

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

$\small w=(1,2)$ $\displaystyle \small w=(1,2)$

find the sum $\small v+w$ $\displaystyle \small v+w$.

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

Given the vectors

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

$\small w=(1,2)$ $\displaystyle \small w=(1,2)$

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

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

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Calculus 3 : Vector Addition

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

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