Vector Subtraction - AP Calculus BC

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Question

Find the difference

Answer

The difference of two vectors and is defined as

For the two vectors

The difference is

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Question

Given the vectors

find .

Answer

Given the vectors

we can find the difference by subtracting component-wise

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Question

Perform the following operation on the vectors specified below:

$3\mathbf{u}-2\mathbf{v}$

Where

$\mathbf{u}=4 \mathbf{i}+8\mathbf{j}+12\mathbf{k},: \mathbf{v}=3\mathbf{i}+6\mathbf{j}+9\mathbf{k}$

Answer

First, multiply out the vectors by their corresponding coefficients:

$3\mathbf{u}=3(4\mathbf{i}+8\mathbf{j}+12\mathbf{k})=12\mathbf{i}+24\mathbf{j}+36\mathbf{k}$

$2\mathbf{v}=2(3\mathbf{i}+6\mathbf{j}+9\mathbf{k})=6\mathbf{i}+12\mathbf{j}+18\mathbf{k}$

Now we simply subtract the two vectors:

$3\mathbf{u}-2\mathbf{v}=(12-6)\mathbf{i}+(24-12)\mathbf{j}+(36-18)\mathbf{k}=6\mathbf{i}+12\mathbf{j}+18\mathbf{k}$

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Question

Subtract the vectors $\left \langle -3,6,-9\right \rangle$ and $\left \langle -1,2,-9\right \rangle$ .

Answer

Set up an expression to subtract the vectors.

$\left \langle -3,6,-9\right \rangle-\left \langle -1,2,-9\right \rangle$

Be sure to make note of the sign change when subtracting the vectors.

$\left \langle -3-(-1),6-2,(-9)-(-9)\right \rangle$

Simplify the vector.

The answer is: $\left \langle -2,4,0\right \rangle$

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Question

Find the difference, , of the two vectors.

Answer

To find the difference, , we distribute the minus sign and turn it into vector addition.

The sum of two vectors is defined as

The sum of two vectors

is defined as

As such, the difference is

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Question

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

Answer

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 $\left \langle (1-2),(3--4),(2-1)\right \rangle=\left \langle -1,7,1\right \rangle$

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Question

Compute the following operation: $\left \langle 2,5,19\right \rangle-\left \langle 3,0,1\right \rangle$ .

Answer

The formula for subtracting vectors a and b is

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

Plugging in the values we were given, we get

$\left \langle (2-3),(5-0),(19-1)\right \rangle=\left \langle -1,5,18\right \rangle$

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Question

Solve:

$\left \langle z, x, y^2z\right \rangle-\left \langle \ln(z), x, 2\right \rangle$

Answer

To subtract vectors, we simply subtract their respective components:

$\left \langle z-\ln(z), 0, y^2z-2\right \rangle$

Note that our result is a vector still.

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Question

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

Answer

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

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

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Question

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

Answer

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 $\left \langle (2-5),(12-5),(13-1) \right \rangle=\left \langle -3,7,12\right \rangle$

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Question

Solve:

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

Answer

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

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

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Question

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

Answer

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

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

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Question

Find the difference between the vectors $\left \langle -10,-20,30\right \rangle$ and $\left \langle 27,15,60\right \rangle$

Answer

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

$\left \langle-10-27,-20-15,30-60\right \rangle=\left \langle-37,-35,-30\right \rangle$

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Question

Find the difference between the vectors $\left \langle 4,0,8\right \rangle$ and $\left \langle 5,-6,2\right \rangle$

Answer

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 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

$\left \langle 4-5,0+6,8-2\right \rangle=\left \langle -1,6,6\right \rangle$

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Question

Find the difference between the vectors $\left \langle 0,-5,4\right \rangle$ and $\left \langle 2,-4,3\right \rangle$

Answer

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 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

$\left \langle 0-2,-5+4,4-3\right \rangle=\left \langle -2,-1,1\right \rangle$

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Question

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

Answer

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

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

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Question

Find the difference between the vectors $\left \langle 5,8,9\right \rangle$ and $\left \langle 3,9,0\right \rangle$

Answer

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

$\left \langle (5-3),(8-9),(9-0) \right \rangle=\left \langle 2,-1,9\right \rangle$

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Question

Find the difference between the vectors $\left \langle \sin(x),2y^2,z^3\right \rangle$ and $\left \langle 2\sin(x),5y^2,-2z^3\right \rangle$

Answer

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

$\left \langle \sin(x)-2\sin(x),2y^2-5y^2,z^3+2z^3\right \rangle=\left \langle -\sin(x),-3y^2,3z^3\right \rangle$

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Question

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

Answer

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

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

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Question

Find the difference between the vectors $\left \langle 3,4,7\right \rangle$ and $\left \langle 3,-3,2\right \rangle$

Answer

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

$\left \langle 3-3,4+3,7-2\right \rangle=\left \langle 0,7,5\right \rangle$

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