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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #71 : Vector Addition

Find the sum of the vectors $\left \langle 2,7,5\right \rangle$ and $\left \langle -2,10,0\right \rangle$

Possible Answers:

$\left \langle 2,17,-4\right \rangle$

$\left \langle 0,17,5\right \rangle$

$\left \langle 1,5,7\right \rangle$

Correct answer:

$\left \langle 0,17,5\right \rangle$

Explanation:

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

$\left \langle 2-2,7+10,5+0\right \rangle=\left \langle 0,17,5\right \rangle$

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Example Question #281 : Vectors And Vector Operations

Solve:

$\left \langle 3, 7, 4\right \rangle +\left \langle x, y^2, z\right \rangle$

Possible Answers:

3x+7y^2+4z

14+x+y^2+z

$\left \langle 3+x, 7+y^2, 4+z\right \rangle$

$\left \langle 3x, 7y^2, 4z\right \rangle$

Correct answer:

$\left \langle 3+x, 7+y^2, 4+z\right \rangle$

Explanation:

To add two vectors together, we simply add the corresponding components (for example, $\left \langle a,b\right \rangle+ \left \langle c,d\right \rangle=\left \langle a+c, b+d\right \rangle$ )

Our final answer is

$\left \langle 3+x, 7+y^2, 4+z\right \rangle$

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Example Question #282 : Vectors And Vector Operations

Find the sum of the vectors $\left \langle 8,6,5\right \rangle$ and $\left \langle -10,10,4\right \rangle$

Possible Answers:

$\left \langle -2,16,9\right \rangle$

$\left \langle -1,16,9\right \rangle$

$\left \langle -2,11,9\right \rangle$

$\left \langle -2,16,7\right \rangle$

Correct answer:

$\left \langle -2,16,9\right \rangle$

Explanation:

To find the sum of 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

$\left \langle 8-10,6+10,5+4\right \rangle= \left \langle -2,16,9\right \rangle$

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Example Question #74 : Vector Addition

Find the vector sum $\mathbf a + \mathbf b$ , where $\mathbf a = \left \langle 3,4,-5 \right \rangle$ and $\mathbf b = \left \langle 2,-2,4 \right \rangle$ .

Possible Answers:

$\left \langle 1,6,-9\right \rangle$

$\left \langle 5,2,-1\right \rangle$

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

$\left \langle 5,-2,1\right \rangle$

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

Correct answer:

$\left \langle 5,2,-1\right \rangle$

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 "sum" of two vectors, is defined by the sum 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 $\mathbf a = \left \langle 3,4,-5 \right \rangle$ and $\mathbf b = \left \langle 2,-2,4 \right \rangle$ . By the definition of the "sum" of two vectors, the vector $\mathbf a + \mathbf b$ can be calculated as follows:

$\mathbf a + \mathbf b=\left \langle (3)+(2), (4)+(-2), (-5)+(4)\right \rangle=\left \langle 5, 2, -1\right \rangle$ .

Hence, $\mathbf a + \mathbf b = \left \langle 5, 2, -1\right \rangle$ .

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Example Question #285 : Vectors And Vector Operations

Find the sum of the vectors $\left \langle -6,-7,3\right \rangle$ and $\left \langle 2,1,-5\right \rangle$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

To find the sum of 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

$\left \langle -6+2,-7+1,3-5\right \rangle= \left \langle -4,-6,-2\right \rangle$

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Example Question #283 : Vectors And Vector Operations

Find the sum of the vectors $\left \langle 3,7,-1\right \rangle$ and $\left \langle 8,2,-6\right \rangle$

Possible Answers:

$\left \langle 11,9,-7\right \rangle$

$\left \langle 12,10,-4\right \rangle$

$\left \langle 11,8,-5\right \rangle$

$\left \langle 10,9,-7\right \rangle$

Correct answer:

$\left \langle 11,9,-7\right \rangle$

Explanation:

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

$\left \langle 3+8,7+2,-1-6\right \rangle=\left \langle 11,9,-7\right \rangle$

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Example Question #76 : Vector Addition

Find the sum of the vectors $\left \langle 4,0,13\right \rangle$ and $\left \langle 5,-7,8\right \rangle$

Possible Answers:

$\left \langle 10,-6,11\right \rangle$

$\left \langle 9,-7,21\right \rangle$

$\left \langle 12,-5,21\right \rangle$

$\left \langle 9,-1,22\right \rangle$

Correct answer:

$\left \langle 9,-7,21\right \rangle$

Explanation:

To find the sum of 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 following 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

$\left \langle 4+5,0-7,13+8\right \rangle=\left \langle 9,-7,21\right \rangle$

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Example Question #282 : Vectors And Vector Operations

Find the difference u-v

u=<2,3>

v=<0,1>

Possible Answers:

<-2,-2>

<2,4>

<2,2>

Correct answer:

<2,2>

Explanation:

The difference a-b of two vectors <a_1,a_2> and <b_1,b_2> is defined as

<a_1-b_1,a_2-b_2>

For the two vectors

u=<2,3>

v=<0,1>

The difference is

u-v=<2,3>-<0,1>

=<2-0,3-1>

=<2,2>

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Example Question #282 : Vectors And Vector Operations

Given the vectors

v=(1,3)

w=(5,2)

find v-w .

Possible Answers:

v-w=(6,5)

v-w=(4,-1)

v-w=(-4,1)

v-w=(1,-4)

Correct answer:

v-w=(-4,1)

Explanation:

Given the vectors

v=(1,3)

w=(5,2)

we can find the difference v-w by subtracting component-wise

v-w=(1,3)-(5,2)=(1-5,3-2)=(-4,1)

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Example Question #3 : Vector Subtraction

Perform the following operation on the vectors specified below:

$3\mathbf{u}-2\mathbf{v}$

Where

$\mathbf{u}=4 \mathbf{i}+8\mathbf{j}+12\mathbf{k},\: \mathbf{v}=3\mathbf{i}+6\mathbf{j}+9\mathbf{k}$

Possible Answers:

$6\mathbf{i}+12\mathbf{j}+18\mathbf{k}$

$17\mathbf{i}+34\mathbf{j}+51\mathbf{k}$

$7\mathbf{i}+12\mathbf{j}+17\mathbf{k}$

$\mathbf{i}+2\mathbf{j}+3\mathbf{k}$

$18\mathbf{i}+36\mathbf{j}+54\mathbf{k}$

Correct answer:

$6\mathbf{i}+12\mathbf{j}+18\mathbf{k}$

Explanation:

First, multiply out the vectors by their corresponding coefficients:

$3\mathbf{u}=3(4\mathbf{i}+8\mathbf{j}+12\mathbf{k})=12\mathbf{i}+24\mathbf{j}+36\mathbf{k}$

$2\mathbf{v}=2(3\mathbf{i}+6\mathbf{j}+9\mathbf{k})=6\mathbf{i}+12\mathbf{j}+18\mathbf{k}$

Now we simply subtract the two vectors:

$3\mathbf{u}-2\mathbf{v}=(12-6)\mathbf{i}+(24-12)\mathbf{j}+(36-18)\mathbf{k}=6\mathbf{i}+12\mathbf{j}+18\mathbf{k}$

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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 #71 : Vector Addition

Example Question #281 : Vectors And Vector Operations

Example Question #282 : Vectors And Vector Operations

Example Question #74 : Vector Addition

Example Question #285 : Vectors And Vector Operations

Example Question #283 : Vectors And Vector Operations

Example Question #76 : Vector Addition

Example Question #282 : Vectors And Vector Operations

Example Question #282 : Vectors And Vector Operations

Example Question #3 : Vector Subtraction

All Calculus 3 Resources

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