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Calculus 2 : Parametric, Polar, and Vector

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Example Questions

Example Question #185 : Vector

Evaluate the dot product: $\left \langle a^2,2a^2\right \rangle \cdot \left \langle 2a^2,a^2\right \rangle$

Possible Answers:

3a^4

$\left \langle 4a^2,4a^2\right \rangle$

$\left \langle 3a^2,3a^2\right \rangle$

2a^4

4a^4

Correct answer:

4a^4

Explanation:

To evaluate the dot product, apply the following formula:

$\left \langle x_1,y_1 \right \rangle \cdot \left \langle x_2,y_2 \right \rangle= x_1x_2+ y_1y_2$

$\left \langle a^2,2a^2\right \rangle \cdot \left \langle 2a^2,a^2\right \rangle= (a^2)(2a^2)+(2a^2)(a^2)= 2a^4+2a^4$

$\left \langle a^2,2a^2\right \rangle \cdot \left \langle 2a^2,a^2\right \rangle=4a^4$

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Example Question #186 : Vector

Evaluate: $\left \langle 1,2,3\right \rangle \times \left \langle 4,5,6\right \rangle$

Possible Answers:

$\left \langle -3,6,-3\right \rangle$

$\left \langle 5,-7,3\right \rangle$

$\left \langle -3,18,-3\right \rangle$

$\left \langle 4,10,18\right \rangle$

Correct answer:

$\left \langle -3,6,-3\right \rangle$

Explanation:

Do not mistaken the times symbol for multiplication. This is a notation for computing the cross product of two vectors.

Write the formula to compute the cross product of two vectors.

For $\vec{x}= \left \langle x_1,x_2,x_3\right \rangle$ and $\vec{y}= \left \langle y_1,y_2,y_3\right \rangle$ :

$\vec{x}\times \vec{y}=\left \langle x_2y_3-x_3y_2, x_3y_1-x_1y_3,x_1y_2-x_2y_1 \right \rangle$

Substitute the values and solve for the cross product.

$\vec{x}\times \vec{y}=\left \langle (2)(6)-(3)(5), (3)(4)-(1)(6), (1)(5)-(2)(4) \right \rangle$

$\vec{x}\times \vec{y}=\left \langle -3,6,-3\right \rangle$

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Example Question #187 : Vector

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

Possible Answers:

None of the above

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of the vectors' composite elements. Thus, given $a=\left \langle 3,7,-10\right \rangle$ and $b=\left \langle 4,6,10 \right \rangle$ .

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

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

=12+42+(-100)

=54+(-100)

=-46

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Example Question #188 : Vector

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

Possible Answers:

$\sqrt{6}$

$\sqrt{2}$

$2\sqrt{2}$

$\sqrt{-1}$

Correct answer:

$\sqrt{6}$

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 $a=\left \langle -2,-1,1\right \rangle$ , then:

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

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

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

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Example Question #189 : Vector

What is the norm of $a=\left \langle 4,0,5\right \rangle$ ?

Possible Answers:

$\sqrt{5}$

$\sqrt{14}$

None of the above

$\sqrt{41}$

Correct answer:

$\sqrt{41}$

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 $a=\left \langle 4,0,5\right \rangle$ , then:

$\left \| a\right \|=\sqrt{(4)^{2}+(0)^{2}+(5)^{2}}$

$\left \| a\right \|=\sqrt{16+0+25}$

$\left \| a\right \|=\sqrt{41}$

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Example Question #190 : Vector

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

Possible Answers:

$\sqrt{2}$

$\sqrt{1}$

$\sqrt{-1}$

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 #191 : Vector

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

Possible Answers:

$\sqrt{107}$

$\sqrt{101}$

$\sqrt{105}$

$\sqrt{110}$

$\sqrt{111}$

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 #192 : Vector

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

Possible Answers:

$2\sqrt{21}$

None of the above

$4\sqrt{21}$

$21\sqrt{2}$

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 #193 : Vector

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

Possible Answers:

$2\sqrt{2}$

$3\sqrt{2}$

$\sqrt{6}$

$4\sqrt{2}$

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 #194 : Vector

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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Calculus 2 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #185 : Vector

Example Question #186 : Vector

Example Question #187 : Vector

Example Question #188 : Vector

Example Question #189 : Vector

Example Question #190 : Vector

Example Question #191 : Vector

Example Question #192 : Vector

Example Question #193 : Vector

Example Question #194 : Vector

All Calculus 2 Resources

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