Vector Addition - AP Calculus BC

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Question

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

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

Answer

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

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Question

Given the vectors

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

Answer

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

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

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Question

Given the vectors

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

Answer

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

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

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Question

Find the sum of the vectors given below

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

Answer

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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Question

Find the sum of the two vectors.

Answer

The sum of two vectors and is defined as

For the vectors

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Question

Given vectors

find .

Answer

To find the sum , we add their components:

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Question

Given the vectors

find the sum .

Answer

To find the sum of the vectors

we add their components:

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Question

Add the following vectors:

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

Where

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

Answer

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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Question

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

Answer

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 .

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

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

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Question

Given the vectors

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

$\small w=(1,2)$

find the sum $\small v+w$ .

Answer

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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Question

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 $\small v+w$ .

Answer

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:

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

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Question

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 $\small v+w$ .

Answer

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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Question

Given the vectors

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

$\small \small \small \small \small \small w=(0,\sqrt{2},3)$

find the sum $\small v+w$

Answer

Given the vectors

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

$\small \small \small \small \small \small w=(0,\sqrt{2},3)$

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

$\small \small \small \small \small \small v+w=(0,1,\pi)+(0,\sqrt{2},3)=(0+0,1+\sqrt{2},\pi+3)$

$\small \small \small \small =(0,1+\sqrt{2},\pi+3)$

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Question

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 $\small v+w$ .

Answer

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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Question

Given the vectors

$\small \small \small \small \small \small \small v=(\ln 2,1/2,e^2)$

$\small \small \small \small \small \small \small \small w=(e,-1/4,e^4)$

find the sum $\small v+w$ .

Answer

Given the vectors

$\small \small \small \small \small \small \small v=(\ln 2,1/2,e^2)$

$\small \small \small \small \small \small \small \small w=(e,-1/4,e^4)$

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

$\small \small \small \small v+w=(\ln 2,1/2,e^2)+(e,-1/4,e^4)=(\ln2+e,1/2-1/4,e^2+e^4)$

$\small \small \small =(\ln 2+e,1/4,e^2+e^4)$

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Question

Find the sum of the vectors $\left \langle \sin(x),5y,z^3\right \rangle$ and $\left \langle 3\sin(x),-4y,0\right \rangle$

Answer

To find the sum of vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we apply the following formula:

$a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle$ . Using the vectors from the problem statement, we get

$\left \langle \sin(x)+3\sin(x),5y-4y,z^3+0\right \rangle=\left \langle4\sin(x),y,z^3 \right \rangle$

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Question

Given the vectors

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

$\small \small \small \small \small \small \small \small \small w=(0,-1,5)$

find the sum $\small v+w$ .

Answer

Given the vectors

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

$\small \small \small \small \small \small \small \small \small w=(0,-1,5)$

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

$\small \small \small \small \small v+w=(1,-1,-2)+(0,-1,5)=(1+0,-1-1,-2+5)$

$\small \small \small \small =(1,-2,3)$

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Question

Given the vectors

$\small \small \small v=(5,5)$

$\small \small \small \small \small \small \small \small \small \small w=(4,2)$

find the sum $\small v+w$ .

Answer

Given the vectors

$\small \small \small v=(5,5)$

$\small \small \small \small \small \small \small \small \small \small w=(4,2)$

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

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

$\small \small \small \small \small =(9,7)$

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Question

Given the vectors

$\small \small \small v=(-1,-5)$

$\small \small \small w=(1,3)$

find the sum $\small v+w$ .

Answer

Given the vectors

$\small \small \small v=(-1,-5)$

$\small \small \small w=(1,3)$

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

$\small \small \small \small \small \small \small v+w=(-1,-5)+(1,3)=(-1+1,-5+3)$

$\small \small \small \small \small =(0,-2)$

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Question

Given the vectors

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

$\small \small \small w=(4,7)$

find the sum $\small v+w$ .

Answer

Given the vectors

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

$\small \small \small w=(4,7)$

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

$\small \small \small \small \small \small \small \small v+w=(2,3)+(4,7)=(2+4,3+7)$

$\small \small \small \small \small =(6,10)$

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