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Calculus 2 : Vector Calculations

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Example Question #31 : Vector Calculations

What is the norm of $a=\left \langle 1,0,-1\right \rangle$ ?

Possible Answers:

$\sqrt{-1}$

$\sqrt{1}$

$\sqrt{2}$

Correct answer:

$\sqrt{2}$

Explanation:

In order to find the norm of a vector, we must take the square root of the sums of the squares of the vector's elements. Given $a=\left \langle 1,0,-1\right \rangle$ , then:

$\left \| a\right \|=\sqrt{(1)^{2}+(0)^{2}+(-1)^{2}}$

$\left \| a\right \|=\sqrt{1+0+1}$

$\left \| a\right \|=\sqrt{2}$

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Example Question #32 : Vector Calculations

What is the norm of $a=\left \langle 5,-6,7\right \rangle$ ?

Possible Answers:

$\sqrt{110}$

$\sqrt{105}$

$\sqrt{101}$

$\sqrt{107}$

$\sqrt{111}$

Correct answer:

$\sqrt{110}$

Explanation:

In order to find the norm of a vector, we must first find the sum of the squares of the vector's elements and take the square root of that sum. Given $a=\left \langle 5,-6,7\right \rangle$ , then:

$\left \| a\right \|=\sqrt{(5)^{2}+(-6)^{2}+(7)^{2}}$

$\left \| a\right \|=\sqrt{25+36+49}$

$\left \| a\right \|=\sqrt{61+49}$

$\left \| a\right \|=\sqrt{110}$

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Example Question #33 : Vector Calculations

What is the norm of $a=\left \langle 2,8,-4\right \rangle$ ?

Possible Answers:

None of the above

$2\sqrt{21}$

$4\sqrt{21}$

$21\sqrt{2}$

Correct answer:

$2\sqrt{21}$

Explanation:

In order to find the norm of a vector, we must first find the sum of the squares of the vector's elements and take the square root of that sum. Given $a=\left \langle 2,8,-4\right \rangle$ , then:

$\left \| a\right \|=\sqrt{(2)^{2}+(8)^{2}+(-4)^{2}}$

$\left \| a\right \|=\sqrt{4+64+16}$

$\left \| a\right \|=\sqrt{4+80}$

$\left \| a\right \|=\sqrt{84}$

$\left \| a\right \|=\sqrt{4\times 21}$

$\left \| a\right \|=2\sqrt{21}$

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Example Question #34 : Vector Calculations

What is the norm of $a=\left \langle 1,-1,-2\right \rangle$ ?

Possible Answers:

$4\sqrt{2}$

$2\sqrt{2}$

$3\sqrt{2}$

$\sqrt{6}$

Correct answer:

$\sqrt{6}$

Explanation:

In order to find the norm of a vector, we must first find the sum of the squares of the vector's elements and take the square root of that sum. Given $a=\left \langle 1,-1,-2\right \rangle$ , then:

$\left \| a\right \|=\sqrt{(1)^{2}+(-1)^{2}+(-2)^{2}}$

$\left \| a\right \|=\sqrt{1+1+4}$

$\left \| a\right \|=\sqrt{2+4}$

$\left \| a\right \|=\sqrt{6}$

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Example Question #35 : Vector Calculations

Calculate the dot product of $a=\left<4,9,-1\right>$ and $b=\left<7,0,-3\right>$ .

Possible Answers:

None of the above

Correct answer:

Explanation:

We can calculate the dot product of $a=\left<4,9,-1\right>$ and $b=\left<7,0,-3\right>$ by finding the sum of the products of both vectors' corresponding elements. Thus:

$\left<4,9,-1\right> \times \left<7,0,-3\right>$

$=(4\times 7)+(9\times 0)+(-1\times-3)$

=28+0+3

=31

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Example Question #36 : Vector Calculations

Calculate the dot product of $a=\left<3,2,-1\right>$ and $b=\left<-1,2,-3\right>$ .

Possible Answers:

$-\frac{1}{4}$

$\frac{1}{4}$

None of the above

Correct answer:

Explanation:

We can calculate the dot product of $a=\left<3,2,-1\right>$ and $b=\left<-1,2,-3\right>$ by finding the sum of the products of both vectors' corresponding elements. Thus:

$\left<3,2,-1\right> \times \left<-1,2,-3\right>$

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

=(-3)+(4)+(3)

=(-3)+(3)+(4)

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Example Question #37 : Vector Calculations

Calculate the dot product of $a=\left<1,-1,0\right>$ and $b=\left<-1,2,-1\right>$ .

Possible Answers:

None of the above

$\frac{1}{3}$

$-\frac{1}{3}$

Correct answer:

Explanation:

We can calculate the dot product of $a=\left<1,-1,0\right>$ and $b=\left<-1,2,-1\right>$ by finding the sum of the products of both vectors' corresponding elements. Thus:

$\left<1,-1,0\right> \times \left<-1,2,-1\right>$

$=(1\times -1)+(-1\times 2)+(0\times -1)$

=-1+(-2)+0

=-3+0

=-3

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Example Question #38 : Vector Calculations

What is the dot product of $a=\left \langle 2,3-7\right \rangle$ and $b=\left \langle -4,1,-2\right \rangle$ ?

Possible Answers:

$\frac{1}{9}$

$-\frac{1}{9}$

None of the above

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of the vectors' corresponding elements. Given $a=\left \langle 2,3-7\right \rangle$ and $b=\left \langle -4,1,-2\right \rangle$ , then:

$\left \langle 2,3-7\right \rangle \times \left \langle -4,1,-2\right \rangle$

$=(2\times -4)+(3\times 1)+(-7\times -2)$

=(-8)+3+14

=(-8)+17

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Example Question #39 : Vector Calculations

What is the dot product of $a=\left \langle 3,-3,2\right \rangle$ and $b=\left \langle 2,3,-3\right \rangle$ ?

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of the vectors' corresponding elements. Given $a=\left \langle 3,-3,2\right \rangle$ and $b=\left \langle 2,3,-3\right \rangle$ , then:

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

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

=6+(-9)+(-6)

=0 +(-9)

=-9

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Example Question #40 : Vector Calculations

What is the dot product of $a=\left \langle 1,2,0\right \rangle$ and $b=\left \langle 3,0,-1\right \rangle$ ?

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of the vectors' corresponding elements. Given $a=\left \langle 1,2,0\right \rangle$ and $b=\left \langle 3,0,-1\right \rangle$ , then:

$\left \langle 1,2,0\right \rangle\times \left \langle 3,0,-1\right \rangle$

$=(1\times 3)+(2\times 0)+(0\times -1)$

=3+0+0

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Calculus 2 : Vector Calculations

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #31 : Vector Calculations

Example Question #32 : Vector Calculations

Example Question #33 : Vector Calculations

Example Question #34 : Vector Calculations

Example Question #35 : Vector Calculations

Example Question #36 : Vector Calculations

Example Question #37 : Vector Calculations

Example Question #38 : Vector Calculations

Example Question #39 : Vector Calculations

Example Question #40 : Vector Calculations

All Calculus 2 Resources

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