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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #311 : Vectors And Vector Operations

Find the vector difference $\mathbf b - \mathbf a$ , where $\mathbf a = \left \langle 10,\sqrt{2},7 \right \rangle$ and $\mathbf b = \left \langle 5,-3,-6 \right \rangle$ .

Possible Answers:

$\left \langle 5,\sqrt{2}+3, 1\right \rangle$

$\left \langle -5,-3-\sqrt{2}, 13\right \rangle$

$\left \langle 5,\sqrt{2}+3, 13\right \rangle$

$\left \langle -5,-3-\sqrt{2}, -13\right \rangle$

$\left \langle -5,3-\sqrt{2}, -13\right \rangle$

Correct answer:

$\left \langle -5,-3-\sqrt{2}, -13\right \rangle$

Explanation:

If $\mathbf a = \left \langle a_1, a_2, a_3, ... , a_n\right \rangle$ and $\mathbf b = \left \langle b_1, b_2, b_3, ... , b_n\right \rangle$ , then the vector $\mathbf a - \mathbf b$ , called the "difference" of two vectors, is defined by the difference of its respective components; that is,

$\mathbf a - \mathbf b=\left \langle a_1-b_1, a_2-b_2, a_3-b_3, ... , a_n-b_n\right \rangle$ .

We have $\mathbf a = \left \langle 10,\sqrt{2},7 \right \rangle$ and $\mathbf b = \left \langle 5,-3,-6 \right \rangle$ . Since subtraction of real numbers is not commutative, and the difference of vectors is defined by the differences of the real numbers which comprise their components, $\mathbf a - \mathbf b \neq \mathbf b - \mathbf a$ unless $\mathbf a = \mathbf b$ . Therefore, we must careful with the order in which the vectors $\mathbf a$ and $\mathbf b$ are subtracted. We can calculate the vector $\mathbf b-\mathbf a$ using the definition of the "difference" of two vectors as follows:

$\mathbf b-\mathbf a =\left \langle (5)-(10), (-3)-(\sqrt{2}), (-6)-(7)\right \rangle$

$=\left \langle -5, -3-\sqrt{2}, -13\right \rangle$

Hence, $\mathbf b - \mathbf a = \left \langle -5, -3-\sqrt{2}, -13\right \rangle$ .

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Example Question #311 : Vectors And Vector Operations

Solve:

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

Possible Answers:

$\left \langle 4, 6, -5\right \rangle$

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

$\left \langle 2, 6, -5\right \rangle$

Correct answer:

$\left \langle 4, 6, -5\right \rangle$

Explanation:

The difference of two vectors is found by subtracting the corresponding components (for instance, $\left \langle a, b\right \rangle - \left \langle c, d\right \rangle= \left \langle a-c, b-d\right \rangle$ )

Our final answer is

$\left \langle 4, 6, -5\right \rangle$

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Example Question #312 : Vectors And Vector Operations

Find the difference of the vectors $\left \langle -7,-2,-5\right \rangle$ and $\left \langle 0,-5,-4\right \rangle$

Possible Answers:

$\left \langle -7,2,-1\right \rangle$

$\left \langle -6,3,-1\right \rangle$

$\left \langle -7,3,-1\right \rangle$

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

Correct answer:

$\left \langle -7,3,-1\right \rangle$

Explanation:

To find the difference 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

$\left \langle -7-0,-2+5,-5+4\right \rangle=\left \langle -7,3,-1\right \rangle$

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Example Question #314 : Vectors And Vector Operations

Find the difference of the vectors $\left \langle 1,-9,4\right \rangle$ and $\left \langle 0,-5,-4\right \rangle$

Possible Answers:

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

$\left \langle 1,-4,8\right \rangle$

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

$\left \langle 2,-4,0\right \rangle$

Correct answer:

$\left \langle 1,-4,8\right \rangle$

Explanation:

To find the difference 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

$\left \langle 1-0,-9+5,4+4\right \rangle=\left \langle 1,-4,8\right \rangle$

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Example Question #315 : Vectors And Vector Operations

Find the difference between the vectors $\left \langle 3x,2y,9z\right \rangle$ and $\left \langle 11x,7y,13z\right \rangle$

Possible Answers:

$\left \langle -8x,-5y,-4z\right \rangle$

$\left \langle -8x,-5y,4z\right \rangle$

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

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

Correct answer:

$\left \langle -8x,-5y,-4z\right \rangle$

Explanation:

To find the difference 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

$\left \langle 3x-11x,2y-7y,9z-13z\right \rangle=\left \langle -8x,-5y,-4z\right \rangle$

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Example Question #1 : Dot Product

Evaluate the dot product between $u=(e^t, t^2, \ln(t))$ , and $v=(2, \ln(t),3)$ .

Possible Answers:

$u\cdot v=2e^t+t^2\ln(t)+3\ln(t)$

$u\cdot v=2+t^2+3$

$u\cdot v=e^t+2\ln(t)$

$u\cdot v=0$

$u\cdot v=2e^t+3\ln(t)$

Correct answer:

$u\cdot v=2e^t+t^2\ln(t)+3\ln(t)$

Explanation:

All we need to do is multiply like components.

$u\cdot v=2e^t+t^2\ln(t)+3\ln(t)$

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Example Question #2 : Dot Product

Evaluate the dot product of u=(1,2,3) , and v=(9,-2,-10) .

Possible Answers:

$u \cdot v =-25$

$u \cdot v =-20$

$u \cdot v =25$

$u \cdot v =0$

Correct answer:

$u \cdot v =-25$

Explanation:

All we need to do is multiply the like components and add them together.

$u\cdot v=(1)(9)+(2)(-2)+(3)(-10)=9-4-30=-25$

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Example Question #1 : Dot Product

Find the dot product of the following vectors:

$\small \bold{v_1}=(1,0,-1)$

$\small \bold{v_2}=(1,1,1)$

Possible Answers:

$\small 1$

$\small -1$

$\small 0$

$\small 2$

Correct answer:

$\small 0$

Explanation:

To find the dot product between two vectors

$\small \small \small \bold{v}=(v_1,v_2,v_3)$

$\small \small \small \bold{w}=(w_1,w_2,w_3)$

we calculate

$\small \bold{v}\cdot\bold{w}=v_1w_1+v_2w_2+v_3w_3$

so for

$\small \bold{v_1}=(1,0,-1)$

$\small \bold{v_2}=(1,1,1)$

we have

$\small \small \small \small \small \bold{v_1}\cdot \bold{v_2}=1\cdot 1+0\cdot 1-1\cdot 1=1-1=0$

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Example Question #2 : Dot Product

What is the length of the vector

$\small \bold{v}=(3,4,5)$ ?

Possible Answers:

$\small 5\sqrt{2}$

$\small 5$

$\small 50$

$\small \small 2\sqrt{5}$

Correct answer:

$\small 5\sqrt{2}$

Explanation:

We can compute the length of a vector by taking the square root of the dot product of $\small \bold{v}$ and $\small \bold{v}$ , so the length of $\small \bold{v}=(3,4,5)$ is:

$\small |\bold{v}|=\sqrt{\bold{v}\cdot \bold{v}}=\sqrt{3^2+4^2+5^2}=\sqrt{9+25+25}$

$\small =\sqrt{50}=\sqrt{25\cdot 2}=5\sqrt{2}$

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Example Question #3 : Dot Product

Find the dot product of the following vectors:

$\small \small \bold{v_1}=(1,2,3)$

$\small \small \bold{v_2}=(1,4,5)$

Possible Answers:

$\small 20$

$\small \sqrt{24}$

$\small 24$

Correct answer:

$\small 24$

Explanation:

To find the dot product between two vectors

$\small \small \small \bold{v}=(v_1,v_2,v_3)$

$\small \small \small \bold{w}=(w_1,w_2,w_3)$

we calculate

$\small \bold{v}\cdot\bold{w}=v_1w_1+v_2w_2+v_3w_3$

so for

$\small \small \bold{v_1}=(1,2,3)$

$\small \small \bold{v_2}=(1,4,5)$

we have

$\small \small \small \small \small \small \bold{v_1}\cdot \bold{v_2}=1\cdot 1+2\cdot 4+3\cdot 5=1+8+15=24$

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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 #311 : Vectors And Vector Operations

Example Question #311 : Vectors And Vector Operations

Example Question #312 : Vectors And Vector Operations

Example Question #314 : Vectors And Vector Operations

Example Question #315 : Vectors And Vector Operations

Example Question #1 : Dot Product

Example Question #2 : Dot Product

Example Question #1 : Dot Product

Example Question #2 : Dot Product

Example Question #3 : Dot Product

All Calculus 3 Resources

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