AP Calculus BC - Vector Addition

Given the vectors

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

$\small w=(1,2)$

find the sum $\small v+w$ .

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

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

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

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

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:

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

Find the sum of the vectors $\left \langle 3,-8,-13\right \rangle$ and $\left \langle 4,0,7\right \rangle$

$\left \langle 7,-8,-6\right \rangle$

$\left \langle 7,8,-8\right \rangle$

$\left \langle 7,6,-8\right \rangle$

$\left \langle 3,5,-8\right \rangle$

Explanation

To find the sum of two 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 use the following formula:

$a+b=\left \langle (x_1+x_2),(y_1+y_2),(z_1+z_2) \right \rangle$

Using the two vectors from the problem statement and applying, we get

$\left \langle (3+4),(-8+0),(-13+7) \right \rangle=\left \langle 7,-8,-6\right \rangle$

Find the sum of the vectors $\left \langle -4,3y^3,z\right \rangle$ and $\left \langle 2,y^3,5z\right \rangle$

$\left \langle -2,4y^3,6z\right \rangle$

$\left \langle -8,3y^6,5z^2\right \rangle$

$\left \langle -2,2y^3,3z\right \rangle$

Explanation

To find the sum of two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ is $a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle$ . Using the vectors we were given, we get $a+b=\left \langle -4+2,3y^3+y^3,z+5z\right \rangle=\left \langle -2,4y^3,6z\right \rangle$

Find the sum of the two vectors.

Explanation

The sum of two vectors

is defined as

For the vectors in this problem

Find the sum of the vectors $\left \langle 3,3y,z^3\right \rangle$ and $\left \langle 5,y,4z^3\right \rangle$

$\left \langle 8,4y,5z^3\right \rangle$

$\left \langle 3,y,4z^3\right \rangle$

$\left \langle 4,2y,4y^3\right \rangle$

Explanation

To find the sum of two vectors $a=\left \langle x_1,y_2,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , you use the 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

$\left \langle (3+5),(3y+y)(z^3+4z^3) \right \rangle=\left \langle 8,4y,5z^3\right \rangle$

Find the sum of the vectors $\left \langle 2x,7y,z\right \rangle$ and $\left \langle 4x,4y,2z\right \rangle$

$\left \langle 6x,11y,3z\right \rangle$

$\left \langle 5x,11y,3z\right \rangle$

$\left \langle 6x,11y,4\right \rangle$

$\left \langle 6x,10y,3z\right \rangle$

Explanation

To find the sum of two 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 use the following formula: