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Calculus 2 : Vector Calculations

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Example Question #11 : Vector Calculations

Find the dot product of $a=\left \langle 3,4,7\right \rangle$ and $b=\left \langle -1,5,6\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

To find the dot product of $a=\left \langle 3,4,7\right \rangle$ and $b=\left \langle -1,5,6\right \rangle$ , calculate the sum of the products of the vectors' corresponding components:

$\left \langle 3,4,7\right \rangle\times \left \langle -1,5,6\right \rangle$

$=(3\times -1)+ (4\times 5)+(7\times 6)$

=-3+ 20+42

=-3+ 62

=59

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Example Question #12 : Vector Calculations

Find the dot product of $a=\left \langle 5,-5,2\right \rangle$ and $b=\left \langle 2,3,8\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

To find the dot product of $a=\left \langle 5,-5,2\right \rangle$ and $b=\left \langle 2,3,8\right \rangle$ , calculate the sum of the products of the vectors' corresponding components:

$\left \langle 5,-5,2\right \rangle\times \left \langle 2,3,8\right \rangle$

$=(5\times 2)+ (-5\times 3)+(2\times 8)$

=10+ (-15)+16

=10+ 1

=11

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Example Question #13 : Vector Calculations

Find the dot product of $a=\left \langle 9,-8,7\right \rangle$ and $b=\left \langle 4,-1,2\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

To find the dot product of $a=\left \langle 9,-8,7\right \rangle$ and $b=\left \langle 4,-1,2\right \rangle$ , calculate the sum of the products of the vectors' corresponding components:

$\left \langle 9,-8,7\right \rangle\times \left \langle 4,-1,2\right \rangle$

$=(9\times 4)+ (-8\times -1)+ (7\times 2)$

=36+ 8+ 14

=36+ 22

=58

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Example Question #14 : Vector Calculations

Find the dot product of $a=\left \langle 2,5,-4\right \rangle$ and $b=\left \langle 1,7,5\right \rangle$

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of their composite elements. Given $a=\left \langle 2,5,-4\right \rangle$ and $b=\left \langle 1,7,5\right \rangle$ , the dot product would therefore be:

$\left \langle 2,5,-4\right \rangle\times \left \langle 1,7,5\right \rangle$

$=(2\times 1)+(5\times 7)+(-4\times 5)$

=2+35+(-20)

=37+(-20)

=17

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Example Question #15 : Vector Calculations

Find the dot product of $a=\left \langle 6,7,-3\right \rangle$ and $b=\left \langle -4,-5,10\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of their composite elements. Given $a=\left \langle 6,7,-3\right \rangle$ and $b=\left \langle -4,-5,10\right \rangle$ , the dot product would therefore be:

$\left \langle 6,7,-3\right \rangle\times \left \langle -4,-5,10\right \rangle$

$=(6\times -4)+(7\times -5)+(-3\times 10)$

=(-24)+(-35)+(-30)

=(-24)+(-65)

=-89

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Example Question #16 : Vector Calculations

Find the dot product of $a=\left \langle 4,2,-2\right \rangle$ and $b=\left \langle 2,3,-6\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of their composite elements. Given $a=\left \langle 4,2,-2\right \rangle$ and $b=\left \langle 2,3,-6\right \rangle$ , the dot product would therefore be:

$\left \langle 4,2,-2\right \rangle\times\left \langle 2,3,-6\right \rangle$

$=(4\times 2)+(2\times 3)+(-2\times -6)$

=(8)+(6)+(12)

=14+12

=26

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Example Question #17 : Vector Calculations

Find the magnitude of $2\vec{a}$ if $\vec{a}= \left \langle -4,3\right \rangle$ .

Possible Answers:

$2\sqrt{73}$

Correct answer:

Explanation:

Evaluate $2\vec{a}$ .

$2\vec{a}=2\left \langle -4,3\right \rangle=\left \langle -8,6\right \rangle$

Find the magnitude.

$\left \| 2\vec{a}\right \|=\sqrt{(-8)^2+6^2} = \sqrt{64+36}= \sqrt{100}=10$

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Example Question #18 : Vector Calculations

Find the dot product of $a=\left \langle 5,-2,3\right \rangle$ and $b=\left \langle -8,1,5\right \rangle$

Possible Answers:

Correct answer:

Explanation:

By definition, the dot product of any two vectors is the sum of the products of their composite elements. Therefore:

$\left \langle 5,-2,3\right \rangle\times \left \langle -8,1,5\right \rangle$

$=(5\times -8)+(-2\times 1)+(3\times 5)$

=(-40)+(-2)+(15)

=-42+15

=-27

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Example Question #11 : Vector Calculations

Find the dot product of $a=\left \langle 4,7,11\right \rangle$ and $b=\left \langle -3,4,-2\right \rangle$

Possible Answers:

None of the above

Correct answer:

Explanation:

By definition, the dot product of any two vectors is the sum of the products of their composite elements. Therefore:

$\left \langle 4,7,11\right \rangle\times \left \langle -3,4,-2\right \rangle$

$=(4\times -3)+(7\times 4)+(11\times -2)$

=(-12)+(28)+(-22)

=16+(-22)

=-6

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Example Question #20 : Vector Calculations

Find the dot product of $a=\left \langle 1,-1,1\right \rangle$ and $b=\left \langle -1,1,-1\right \rangle$

Possible Answers:

Correct answer:

Explanation:

By definition, the dot product of any two vectors is the sum of the products of their composite elements. Therefore:

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

$=(1\times -1)+(-1\times 1)+(1\times -1)$

=(-1)+(-1)+(-1)

=-3

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Calculus 2 : Vector Calculations

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #11 : Vector Calculations

Example Question #12 : Vector Calculations

Example Question #13 : Vector Calculations

Example Question #14 : Vector Calculations

Example Question #15 : Vector Calculations

Example Question #16 : Vector Calculations

Example Question #17 : Vector Calculations

Example Question #18 : Vector Calculations

Example Question #11 : Vector Calculations

Example Question #20 : Vector Calculations

All Calculus 2 Resources

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