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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 #61 : Vector Addition

Find the sum of the vectors $\left \langle 3,5,9\right \rangle$ and $\left \langle 2,0,1\right \rangle$

Possible Answers:

$\left \langle 5,5,10\right \rangle$

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

$\left \langle 0,5,10\right \rangle$

$\left \langle 4,5,11\right \rangle$

Correct answer:

$\left \langle 5,5,10\right \rangle$

Explanation:

To find the sum 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

$\left \langle3+2,5+0,9+1\right \rangle=\left \langle 5,5,10\right \rangle$

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Example Question #62 : Vector Addition

Find the sum of the vectors $\left \langle 3,-1,-5\right \rangle$ and $\left \langle -7,6,3\right \rangle$

Possible Answers:

$\left \langle -4,5,-2\right \rangle$

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

$\left \langle -4,-5,-7\right \rangle$

$\left \langle -4,-5,-2\right \rangle$

Correct answer:

$\left \langle -4,-5,-2\right \rangle$

Explanation:

To find the sum 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

$\left \langle 3-7,-1+6,-5+3\right \rangle=\left \langle -4,-5,-2\right \rangle$

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Example Question #63 : Vector Addition

Find the sum of the vectors $\left \langle 3,-5,7\right \rangle$ and $\left \langle -2,1,3\right \rangle$

Possible Answers:

$\left \langle 1,-3,10\right \rangle$

$\left \langle 2,-4,10\right \rangle$

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

$\left \langle 1,-4,10\right \rangle$

Correct answer:

$\left \langle 1,-4,10\right \rangle$

Explanation:

To find the sum 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

$\left \langle 3-2,-5+1,7+3\right \rangle=\left \langle 1,-4,10\right \rangle$

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Example Question #64 : Vector Addition

Find the sum of the vectors $\left \langle 6,8,2\right \rangle$ and $\left \langle 4,10,13\right \rangle$

Possible Answers:

$\left \langle 10,16,15\right \rangle$

$\left \langle 11,18,15\right \rangle$

$\left \langle 10,18,15\right \rangle$

$\left \langle 10,18,14\right \rangle$

Correct answer:

$\left \langle 10,18,15\right \rangle$

Explanation:

To find the sum 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

$\left \langle 6+4,8+10,2+13\right \rangle=\left \langle 10,18,15\right \rangle$

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Example Question #65 : Vector Addition

Find the sum of the vectors $\left \langle 3,2y,12\right \rangle$ and $\left \langle 7,5y,6\right \rangle$

Possible Answers:

$\left \langle 10,8y,16\right \rangle$

$\left \langle 10,7y,18\right \rangle$

$\left \langle 11,8,18\right \rangle$

$\left \langle 9,8y,18\right \rangle$

Correct answer:

$\left \langle 10,7y,18\right \rangle$

Explanation:

To find the sum 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

$\left \langle 3+7,2y+5y,12+6\right \rangle=\left \langle 10,7y,18\right \rangle$

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Example Question #66 : Vector Addition

Find the sum of the vectors $\left \langle 5,0,3\right \rangle$ and $\left \langle 1,-1,3\right \rangle$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

To find the sum 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

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

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Example Question #67 : Vector Addition

Solve:

$\left \langle 3+x, 5, x+1 \right\rangle + \left \langle x^2, 2, 5\right \rangle$

Possible Answers:

$\left \langle 3x^2+x^3, 10, 5x+5\right \rangle$

$\left \langle x^2+x+3, 6, x+6\right \rangle$

$\left \langle x^2+x+3, 7, x+6\right \rangle$

x^2+2x+16

Correct answer:

$\left \langle x^2+x+3, 7, x+6\right \rangle$

Explanation:

To add two vectors together, we simply add 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

$\left \langle x^2+x+3, 7, x+6\right \rangle$

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Example Question #71 : Vector Addition

Solve:

$\left \langle xy, y^2, 1\right \rangle+\left \langle z, 2, 4\right \rangle$

Possible Answers:

xy+y^2+z+7

$\left \langle xy+z, 2+y^2, 5\right \rangle$

$\left \langle xy+z, 2+y^2, 4\right \rangle$

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

Correct answer:

$\left \langle xy+z, 2+y^2, 5\right \rangle$

Explanation:

The sum of two vectors found by adding 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

$\left \langle xy+z, 2+y^2, 5\right \rangle$

(Note that the sum of two vectors produces another vector.)

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Example Question #72 : Vector Addition

Solve:

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

Possible Answers:

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

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

x^2+y+z+4

$\left \langle 3+x^2, y+z, 1\right \rangle$

Correct answer:

$\left \langle 3+x^2, y+z, 1\right \rangle$

Explanation:

To add vectors, we add their corresponding components together (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

$\left \langle 3+x^2, y+z, 1\right \rangle$

after using a Pythagorean identity to simplify.

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Example Question #73 : Vector Addition

Find the sum of the vectors $\left \langle 9,1,8\right \rangle$ and $\left \langle -1,-6,-4\right \rangle$

Possible Answers:

$\left \langle 8,-5,2\right \rangle$

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

$\left \langle 8,-5,4\right \rangle$

Correct answer:

$\left \langle 8,-5,4\right \rangle$

Explanation:

To find the sum of two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you use the following 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

$\left \langle 9-1,1-6,8-4\right \rangle=\left \langle 8,-5,4\right \rangle$

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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 #61 : Vector Addition

Example Question #62 : Vector Addition

Example Question #63 : Vector Addition

Example Question #64 : Vector Addition

Example Question #65 : Vector Addition

Example Question #66 : Vector Addition

Example Question #67 : Vector Addition

Example Question #71 : Vector Addition

Example Question #72 : Vector Addition

Example Question #73 : Vector Addition

All Calculus 3 Resources

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