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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #12 : Vector Subtraction

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

Possible Answers:

$\left \langle -1,5,6\right \rangle$

$\left \langle -1,6,6\right \rangle$

$\left \langle -2,6,6\right \rangle$

$\left \langle -1,-6,6\right \rangle$

Correct answer:

$\left \langle -1,6,6\right \rangle$

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 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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Example Question #12 : Vector Subtraction

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

Possible Answers:

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

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

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

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

Correct answer:

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

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 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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Example Question #301 : Vectors And Vector Operations

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

Possible Answers:

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

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

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

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

Correct answer:

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

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

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

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Example Question #13 : Vector Subtraction

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

Possible Answers:

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

$\left \langle 2,-1,8\right \rangle$

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

$\left \langle 2,1,9\right \rangle$

Correct answer:

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

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

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

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Example Question #14 : Vector Subtraction

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

Possible Answers:

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

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

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

$\left \langle -\sin(x),-3y^2,4z^3\right \rangle$

Correct answer:

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

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

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

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Example Question #13 : Vector Subtraction

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

Possible Answers:

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

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

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

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

Correct answer:

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

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

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

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Example Question #301 : Vectors And Vector Operations

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

Possible Answers:

$\left \langle 0,7,5\right \rangle$

$\left \langle 2,7,5\right \rangle$

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

$\left \langle 0,7,4\right \rangle$

Correct answer:

$\left \langle 0,7,5\right \rangle$

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

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

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Example Question #301 : Vectors And Vector Operations

Find the difference between the vectors $\left \langle 2,-3,6\right \rangle$ and $\left \langle -4,3,7\right \rangle$

Possible Answers:

$\left \langle 6,-5,-1\right \rangle$

$\left \langle 6,0,-1\right \rangle$

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

$\left \langle 6,-6,-1\right \rangle$

Correct answer:

$\left \langle 6,-6,-1\right \rangle$

Explanation:

To find the difference between 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

$\left \langle 2+4,-3-3,6-7\right \rangle=\left \langle6,-6,-1 \right \rangle$

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Example Question #21 : Vector Subtraction

Solve:

$\left \langle 3, 0, 12\right \rangle - \left \langle 2, -4, 3\right \rangle$

Possible Answers:

$\left \langle 1, -4, 9\right \rangle$

$\left \langle 1, 4, 9\right \rangle$

Correct answer:

$\left \langle 1, 4, 9\right \rangle$

Explanation:

To subtract two vectors, we must take the difference between the corresponding components (for example, $\left \langle a, b\right \rangle - \left \langle c, d\right \rangle= \left \langle a-c, b-d\right \rangle$ )

Our final answer is

$\left \langle 1, 4, 9\right \rangle$

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Example Question #22 : Vector Subtraction

Find the difference between the vectors $\left \langle 3,3,4\right \rangle$ and $\left \langle -1,2,5\right \rangle$

Possible Answers:

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

$\left \langle 4,1,6\right \rangle$

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

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

Correct answer:

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

Explanation:

To find the difference between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you 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

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

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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #12 : Vector Subtraction

Example Question #12 : Vector Subtraction

Example Question #301 : Vectors And Vector Operations

Example Question #13 : Vector Subtraction

Example Question #14 : Vector Subtraction

Example Question #13 : Vector Subtraction

Example Question #301 : Vectors And Vector Operations

Example Question #301 : Vectors And Vector Operations

Example Question #21 : Vector Subtraction

Example Question #22 : Vector Subtraction

All Calculus 3 Resources

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