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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #2261 : Calculus 3

Find the sum of the vectors $\left \langle -3,-9,1\right \rangle$ $\displaystyle \left \langle -3,-9,1\right \rangle$ and $\left \langle 15,10,4\right \rangle$ $\displaystyle \left \langle 15,10,4\right \rangle$

Possible Answers:

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

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

$\left \langle 12,1,5\right \rangle$ $\displaystyle \left \langle 12,1,5\right \rangle$

$\left \langle 11,1,5\right \rangle$ $\displaystyle \left \langle 11,1,5\right \rangle$

Correct answer:

$\left \langle 12,1,5\right \rangle$ $\displaystyle \left \langle 12,1,5\right \rangle$

Explanation:

To find the sum between vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$, we use the following formula:

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

Using the vectors from the problem statement, we get

$\left \langle (-3+15),(-9+10),(1+4) \right \rangle=\left \langle 12,1,5\right \rangle$ $\displaystyle \left \langle (-3+15),(-9+10),(1+4) \right \rangle=\left \langle 12,1,5\right \rangle$

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Example Question #2262 : Calculus 3

Find the sum of the vectors $\left \langle -9,-5,3\right \rangle$ $\displaystyle \left \langle -9,-5,3\right \rangle$ and $\left \langle 2,0,-8\right \rangle$ $\displaystyle \left \langle 2,0,-8\right \rangle$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

Applying to the vectors from the problem statement, we get

$\left \langle -9+2,-5+0,3-8\right \rangle=\left \langle -7,-5,-5\right \rangle$ $\displaystyle \left \langle -9+2,-5+0,3-8\right \rangle=\left \langle -7,-5,-5\right \rangle$

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Example Question #56 : Vector Addition

Find the sum of the vectors $\left \langle -10,-13,15\right \rangle$ $\displaystyle \left \langle -10,-13,15\right \rangle$ and $\left \langle 5,2,-9\right \rangle$ $\displaystyle \left \langle 5,2,-9\right \rangle$

Possible Answers:

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

$\left \langle 5,-11,6\right \rangle$ $\displaystyle \left \langle 5,-11,6\right \rangle$

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

$\left \langle -5,-11,6\right \rangle$ $\displaystyle \left \langle -5,-11,6\right \rangle$

Correct answer:

$\left \langle -5,-11,6\right \rangle$ $\displaystyle \left \langle -5,-11,6\right \rangle$

Explanation:

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

Applying to the vectors from the problem statement, we get

$\left \langle -10+5,-13+2,15-9\right \rangle=\left \langle -5,-11,6\right \rangle$ $\displaystyle \left \langle -10+5,-13+2,15-9\right \rangle=\left \langle -5,-11,6\right \rangle$

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Example Question #2261 : Calculus 3

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

Possible Answers:

$\left \langle 12,-2,4\right \rangle$ $\displaystyle \left \langle 12,-2,4\right \rangle$

$\left \langle 13,-2,4\right \rangle$ $\displaystyle \left \langle 13,-2,4\right \rangle$

$\left \langle 13,-2,6\right \rangle$ $\displaystyle \left \langle 13,-2,6\right \rangle$

$\left \langle 13,-4,4\right \rangle$ $\displaystyle \left \langle 13,-4,4\right \rangle$

Correct answer:

$\left \langle 13,-2,4\right \rangle$ $\displaystyle \left \langle 13,-2,4\right \rangle$

Explanation:

To find the sum of three vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$, $b=\left \langle x_2,y_2,z_2\right \rangle$ $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$, and $c=\left \langle x_3,y_3,z_3\right \rangle$ $\displaystyle c=\left \langle x_3,y_3,z_3\right \rangle$, we use the following formula:

$a+b+c=\left \langle (x_1+x_2+x_3),(y_1+y_2+y_3),(z_1+z_2+z_3) \right \rangle$ $\displaystyle a+b+c=\left \langle (x_1+x_2+x_3),(y_1+y_2+y_3),(z_1+z_2+z_3) \right \rangle$

Using the two vectors from the problem statement and applying, we get

$\left \langle (3+9+1),(-2+0+0),(1+2+1) \right \rangle=\left \langle 13,-2,4\right \rangle$ $\displaystyle \left \langle (3+9+1),(-2+0+0),(1+2+1) \right \rangle=\left \langle 13,-2,4\right \rangle$

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Example Question #58 : Vector Addition

Find the sum of the vectors $\left \langle 3,-8,-13\right \rangle$ $\displaystyle \left \langle 3,-8,-13\right \rangle$ and $\left \langle 4,0,7\right \rangle$ $\displaystyle \left \langle 4,0,7\right \rangle$

Possible Answers:

$\left \langle 3,5,-8\right \rangle$ $\displaystyle \left \langle 3,5,-8\right \rangle$

$\left \langle 7,6,-8\right \rangle$ $\displaystyle \left \langle 7,6,-8\right \rangle$

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

$\left \langle 7,-8,-6\right \rangle$ $\displaystyle \left \langle 7,-8,-6\right \rangle$

Correct answer:

$\left \langle 7,-8,-6\right \rangle$ $\displaystyle \left \langle 7,-8,-6\right \rangle$

Explanation:

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

Using the two vectors from the problem statement and applying, we get

$\left \langle (3+4),(-8+0),(-13+7) \right \rangle=\left \langle 7,-8,-6\right \rangle$ $\displaystyle \left \langle (3+4),(-8+0),(-13+7) \right \rangle=\left \langle 7,-8,-6\right \rangle$

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Example Question #59 : Vector Addition

Find the sum of the vectors $\left \langle 5,-18,3\right \rangle$ $\displaystyle \left \langle 5,-18,3\right \rangle$ and $\left \langle 3,-6,5\right \rangle$ $\displaystyle \left \langle 3,-6,5\right \rangle$

Possible Answers:

$\left \langle 8,-24,8\right \rangle$ $\displaystyle \left \langle 8,-24,8\right \rangle$

$\left \langle 8,-14,8\right \rangle$ $\displaystyle \left \langle 8,-14,8\right \rangle$

$\left \langle 6,-24,8\right \rangle$ $\displaystyle \left \langle 6,-24,8\right \rangle$

$\left \langle 8,10,20\right \rangle$ $\displaystyle \left \langle 8,10,20\right \rangle$

Correct answer:

$\left \langle 8,-24,8\right \rangle$ $\displaystyle \left \langle 8,-24,8\right \rangle$

Explanation:

To find the sum of two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ $\displaystyle a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ $\displaystyle b=\left \langle x_2,y_2,z_2\right \rangle$, we apply the following formula:
$a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle$ $\displaystyle a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle$

Using the vectors from the problem statement, we get

$\left \langle 5+3,-18-6,3+5\right \rangle=\left \langle 8,-24,8\right \rangle$ $\displaystyle \left \langle 5+3,-18-6,3+5\right \rangle=\left \langle 8,-24,8\right \rangle$

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Example Question #60 : Vector Addition

Find the sum of the vectors $\left \langle 7,-10,4\right \rangle$ $\displaystyle \left \langle 7,-10,4\right \rangle$ and $\left \langle 2,-6,9\right \rangle$ $\displaystyle \left \langle 2,-6,9\right \rangle$

Possible Answers:

$\left \langle 9,-16,11\right \rangle$ $\displaystyle \left \langle 9,-16,11\right \rangle$

$\left \langle 9,-16,13\right \rangle$ $\displaystyle \left \langle 9,-16,13\right \rangle$

$\left \langle 9,-9,13\right \rangle$ $\displaystyle \left \langle 9,-9,13\right \rangle$

$\left \langle 8,-16,13\right \rangle$ $\displaystyle \left \langle 8,-16,13\right \rangle$

Correct answer:

$\left \langle 9,-16,13\right \rangle$ $\displaystyle \left \langle 9,-16,13\right \rangle$

Explanation:

Using the vectors from the problem statement, we get

$\left \langle 7+2,-10-6,4+9\right \rangle=\left \langle 9,-16,13\right \rangle$ $\displaystyle \left \langle 7+2,-10-6,4+9\right \rangle=\left \langle 9,-16,13\right \rangle$

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Example Question #2261 : Calculus 3

Find the sum of the vectors $\left \langle 2x,7y,z\right \rangle$ $\displaystyle \left \langle 2x,7y,z\right \rangle$ and $\left \langle 4x,4y,2z\right \rangle$ $\displaystyle \left \langle 4x,4y,2z\right \rangle$

Possible Answers:

$\left \langle 6x,11y,4\right \rangle$ $\displaystyle \left \langle 6x,11y,4\right \rangle$

$\left \langle 5x,11y,3z\right \rangle$ $\displaystyle \left \langle 5x,11y,3z\right \rangle$

$\left \langle 6x,10y,3z\right \rangle$ $\displaystyle \left \langle 6x,10y,3z\right \rangle$

$\left \langle 6x,11y,3z\right \rangle$ $\displaystyle \left \langle 6x,11y,3z\right \rangle$

Correct answer:

$\left \langle 6x,11y,3z\right \rangle$ $\displaystyle \left \langle 6x,11y,3z\right \rangle$

Explanation:

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

Applying to the vectors from the problem statement, we get

$\left \langle (2x+4x),(7y+4y),(z+2z) \right \rangle=\left \langle 6x,11y,3z\right \rangle$ $\displaystyle \left \langle (2x+4x),(7y+4y),(z+2z) \right \rangle=\left \langle 6x,11y,3z\right \rangle$

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Example Question #261 : Vectors And Vector Operations

Find the sum of the vectors $\left \langle x,8y,3z\right \rangle$ $\displaystyle \left \langle x,8y,3z\right \rangle$ and $\left \langle x,4y,5z\right \rangle$ $\displaystyle \left \langle x,4y,5z\right \rangle$

Possible Answers:

$\left \langle 2x,12y,8z\right \rangle$ $\displaystyle \left \langle 2x,12y,8z\right \rangle$

$\left \langle 2x,10y,8z\right \rangle$ $\displaystyle \left \langle 2x,10y,8z\right \rangle$

$\left \langle 2x,12y,9z\right \rangle$ $\displaystyle \left \langle 2x,12y,9z\right \rangle$

$\left \langle 2x,2y,8z\right \rangle$ $\displaystyle \left \langle 2x,2y,8z\right \rangle$

Correct answer:

$\left \langle 2x,12y,8z\right \rangle$ $\displaystyle \left \langle 2x,12y,8z\right \rangle$

Explanation:

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

Applying to the vectors from the problem statement, we get

$\left \langle (x+x),(8y+4y),(3z+5z) \right \rangle=\left \langle 2x,12y,8z\right \rangle$ $\displaystyle \left \langle (x+x),(8y+4y),(3z+5z) \right \rangle=\left \langle 2x,12y,8z\right \rangle$

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Example Question #2272 : Calculus 3

Find the sum of the vectors $\left \langle 2x,5y,z\right \rangle$ $\displaystyle \left \langle 2x,5y,z\right \rangle$ and $\left \langle 10x,9y,4z\right \rangle$ $\displaystyle \left \langle 10x,9y,4z\right \rangle$

Possible Answers:

$\left \langle 12x,10y,6z\right \rangle$ $\displaystyle \left \langle 12x,10y,6z\right \rangle$

$\left \langle 12x,14y,5z\right \rangle$ $\displaystyle \left \langle 12x,14y,5z\right \rangle$

$\left \langle 11x,14y,5z\right \rangle$ $\displaystyle \left \langle 11x,14y,5z\right \rangle$

$\left \langle 2x,11y,8z\right \rangle$ $\displaystyle \left \langle 2x,11y,8z\right \rangle$

Correct answer:

$\left \langle 12x,14y,5z\right \rangle$ $\displaystyle \left \langle 12x,14y,5z\right \rangle$

Explanation:

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

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

Using the vectors from the problem statement, we get

$\left \langle 2x+10x,5y+9y,z+4z\right \rangle=\left \langle 12x,14y,5z\right \rangle$ $\displaystyle \left \langle 2x+10x,5y+9y,z+4z\right \rangle=\left \langle 12x,14y,5z\right \rangle$

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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 #2261 : Calculus 3

Example Question #2262 : Calculus 3

Example Question #56 : Vector Addition

Example Question #2261 : Calculus 3

Example Question #58 : Vector Addition

Example Question #59 : Vector Addition

Example Question #60 : Vector Addition

Example Question #2261 : Calculus 3

Example Question #261 : Vectors And Vector Operations

Example Question #2272 : Calculus 3

All Calculus 3 Resources

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