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Calculus 3 : Vectors and Vector Operations

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #44 : Vector Addition

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

Possible Answers:

8+4y+5z^3

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

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

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

Correct answer:

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

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

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

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Example Question #45 : Vector Addition

Find the sum of the vectors $\left \langle 3,0,8\right \rangle$ and $\left \langle 12,17,25\right \rangle$

Possible Answers:

$\left \langle 15,17,33\right \rangle$

$\left \langle 5,13,25\right \rangle$

$\left \langle -9,-17,-17\right \rangle$

$\left \langle 19,29,33\right \rangle$

Correct answer:

$\left \langle 15,17,33\right \rangle$

Explanation:

$\left \langle (3+12),(0+17)(8+25) \right \rangle=\left \langle 15,17,33\right \rangle$

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Example Question #46 : Vector Addition

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

Possible Answers:

$\left \langle 3,7y,4z^2\right \rangle$

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

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

$\left \langle 0,3y,5z^2\right \rangle$

Correct answer:

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

Explanation:

To find the difference 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

$\left \langle (3-5),(3y-4y)(z^2-4z^3) \right \rangle=\left \langle -2,-y,-3z^2\right \rangle$

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Example Question #47 : Vector Addition

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

Possible Answers:

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

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

$\left \langle 4\cos(x),3y,z^3\right \rangle$

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

Correct answer:

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

Explanation:

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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Example Question #48 : Vector Addition

Find the sum of the vectors $\left \langle4,-9,10 \right \rangle$ and $\left \langle 3,y^2,z\right \rangle$

Possible Answers:

$\left \langle 5,-9y^2,10z\right \rangle$

$\left \langle 3,9y^2,-10z\right \rangle$

$\left \langle 7,-9+y^2,10+z\right \rangle$

$\left \langle 12,-9y^2,10+z\right \rangle$

Correct answer:

$\left \langle 7,-9+y^2,10+z\right \rangle$

Explanation:

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

$\left \langle4+3,-9+y^2,10+z\right \rangle=\left \langle7,-9+y^2,10+z \right \rangle$

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Example Question #49 : Vector Addition

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

Possible Answers:

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

$\left \langle 6x,3\sin(x)+\cos(y),z^3\right \rangle$

$\left \langle 6x,3\sin(y)+\cos(y),4z^3\right \rangle$

$\left \langle 6x^2,3\sin(y)+\cos(y),3z^3\right \rangle$

Correct answer:

$\left \langle 6x,3\sin(y)+\cos(y),4z^3\right \rangle$

Explanation:

To find the sum of the vectors $a=\left \langle x_2,y_1,z_1\right \rangle$ 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 in the problem statement, we get:

$\left \langle x+5x,3\sin(y)+\cos(y),z^3+3z^3\right \rangle=\left \langle 6x,3\sin(y)+\cos(y),4z^3\right \rangle$

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Example Question #50 : Vector Addition

Find the sum of the vectors $\left \langle 4,0,9\right \rangle$ and $\left \langle 3,-8,7\right \rangle$

Possible Answers:

$\left \langle 12,0,63\right \rangle$

$\left \langle 7,-8,16\right \rangle$

$\left \langle 5,-4,1\right \rangle$

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

Correct answer:

$\left \langle 7,-8,16\right \rangle$

Explanation:

To find the sum of the vectors $a=\left \langle x_2,y_1,z_1\right \rangle$ 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 in the problem statement, we get:

$\left \langle 4+3,0-8,9+7\right \rangle=\left \langle 7,-8,16\right \rangle$

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Example Question #51 : Vector Addition

Find the sum of the vectors $\left \langle 3x,y,-2z\right \rangle$ and $\left \langle 6x,2y,0\right \rangle$

Possible Answers:

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

$\left \langle 18x,2y^2,0\right \rangle$

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

$\left \langle 9x,3y,-2z\right \rangle$

Correct answer:

$\left \langle 9x,3y,-2z\right \rangle$

Explanation:

To find the sum 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

$\left \langle 3x+6x,y+2y,-2z+0\right \rangle=\left \langle 9x,3y,-2z\right \rangle$

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Example Question #52 : Vector Addition

Find the sum of the vectors $\left \langle -4x,5y,z\right \rangle$ and $\left \langle 6,7y,1\right \rangle$

Possible Answers:

$\left \langle -4x+6,12y,z+1\right \rangle$

$\left \langle-24x,35y,z \right \rangle$

$\left \langle 4-3x,10,7z\right \rangle$

$\left \langle 17x,12y,2z\right \rangle$

Correct answer:

$\left \langle -4x+6,12y,z+1\right \rangle$

Explanation:

To find the sum 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

$\left \langle -4x+6,5y+7y,z+1\right \rangle=\left \langle -4x+6,12y,z+1\right \rangle$

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Example Question #53 : Vector Addition

Find the sum of the vectors $\left \langle 3,1,2z\right \rangle$ and $\left \langle 9,5,4z\right \rangle$

Possible Answers:

$\left \langle 12,6,6z\right \rangle$

$\left \langle 12,5,6z\right \rangle$

$\left \langle 10,7,6z\right \rangle$

$\left \langle 12,4,z\right \rangle$

Correct answer:

$\left \langle 12,6,6z\right \rangle$

Explanation:

To find the sum between 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

$\left \langle (3+9),(1+5),(2z+4z) \right \rangle=\left \langle 12,6,6z\right \rangle$

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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 #44 : Vector Addition

Example Question #45 : Vector Addition

Example Question #46 : Vector Addition

Example Question #47 : Vector Addition

Example Question #48 : Vector Addition

Example Question #49 : Vector Addition

Example Question #50 : Vector Addition

Example Question #51 : Vector Addition

Example Question #52 : Vector Addition

Example Question #53 : Vector Addition

All Calculus 3 Resources

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