AP Calculus BC - Vector Subtraction

Find the difference

Explanation

The difference of two vectors and is defined as

For the two vectors

The difference is

Find the difference between the vectors $\left \langle -9,0,5\right \rangle$ and $\left \langle 2,2,2\right \rangle$

$\left \langle -11,-2,3\right \rangle$

$\left \langle -11,-2,4\right \rangle$

$\left \langle -11,2,3\right \rangle$

$\left \langle -10,-2,3\right \rangle$

Explanation

To find the difference of two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ we use the formula:

$a+b=\left \langle a_1-b_1,a_2-b_2,a_3-b_3\right \rangle$

Using the vectors from the problem statement, we get

$\left \langle -9-2,0-2,5-2\right \rangle=\left \langle -11,-2,3\right \rangle$

Find the difference between the vectors $\left \langle 3x,8y,4z\right \rangle$ and $\left \langle -2x,y,2z\right \rangle$

$\left \langle 5x,7y,2z\right \rangle$

$\left \langle 5x,7y,z\right \rangle$

$\left \langle 5x,9y,2z\right \rangle$

$\left \langle 7x,7y,2z\right \rangle$

Explanation

To find the difference between 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:

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

Applying to the vectors from the problem statement, we get

$\left \langle (3x+2x),(8y-y),(4z-2z) \right \rangle=\left \langle 5x,7y,2z\right \rangle$

Find the difference between the vectors $\left \langle 4x,7y,-4z\right \rangle$ and $\left \langle 2x,0,-z\right \rangle$

$\left \langle 2x,7y,-3z\right \rangle$

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

$\left \langle 2x,7y,-5z\right \rangle$

$\left \langle -x,7y,-3z\right \rangle$

Explanation

To find the difference between the 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 vectors from the problem statement, we get

$\left \langle 4x-2x,7y-0,-4z+z\right \rangle=\left \langle2x,7y,-3z\right \rangle$

Given the vectors

find .

Explanation

Given the vectors

we can find the difference by subtracting component-wise

Solve:

$\left \langle xy, z\right \rangle- \left \langle 3xy, z\right \rangle$

$\left \langle -2xy, 0\right \rangle$

$\left \langle xy, 0\right \rangle$

$\left \langle -2xy, -2z\right \rangle$

Explanation

The difference between two vectors is given by subtracting the corresponding components (for example, $\left \langle a, b\right \rangle - \left \langle x,y\right \rangle=\left \langle a-x, b-y\right \rangle$ )

Our final answer is

$\left \langle -2xy, 0\right \rangle$

Perform the following vector operation: $\left \langle 2,12,13\right \rangle-\left \langle 5,5,1\right \rangle$

$\left \langle -3,7,12\right \rangle$

$\left \langle 3,7,12\right \rangle$

$\left \langle -2,1,5\right \rangle$

Explanation

The formula for the difference of vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ is $a-b=\left \langle (x_1-x_2),(y_1-y_2),(z_1-z_2) \right \rangle$ . Using the vectors from the problem statement, we get $\left \langle (2-5),(12-5),(13-1) \right \rangle=\left \langle -3,7,12\right \rangle$

Find the difference between $\left \langle 2,0,6\right \rangle$ and $\left \langle 3,3,1\right \rangle$ .

$\left \langle -1,-3,5\right \rangle$

$\left \langle 1,3,1\right \rangle$

$\left \langle 2,0,5\right \rangle$

$\left \langle 0,-1,4\right \rangle$

Explanation

The formula for the difference of vectors is

$\left \langle x_1,x_2,x_3\right \rangle-\left \langle y_1,y_2,y_3\right \rangle=\left \langle x_1-y_1,x_2-y_2,x_3-y_3\right \rangle$

Using the vectors given, we get

$\left \langle 2-3,0-3,6-1\right \rangle=\left \langle -1,-3,5\right \rangle$ .

Compute the following: $\left \langle 1,3,2\right \rangle-\left \langle 2,-4,1\right \rangle$ .

$\left \langle -1,7,1\right \rangle$

$\left \langle -1,1,1\right \rangle$

$\left \langle -2,0,1\right \rangle$

$\left \langle 3,-7,2\right \rangle$

Explanation

The formula for subtracting vectors is $a-b=\left \langle a_1-b_!,a_2-b_2,a_3-b_3\right \rangle$ . Plugging in the values we were given, we get $\left \langle (1-2),(3--4),(2-1)\right \rangle=\left \langle -1,7,1\right \rangle$