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Calculus 3 : Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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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Example Question #3 : Vector Subtraction

Subtract the vectors $\left \langle -3,6,-9\right \rangle$ and $\left \langle -1,2,-9\right \rangle$ .

Possible Answers:

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

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

$\left \langle -2,4,-18\right \rangle$

$\left \langle -4,8,-18\right \rangle$

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

Correct answer:

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

Explanation:

Set up an expression to subtract the vectors.

$\left \langle -3,6,-9\right \rangle-\left \langle -1,2,-9\right \rangle$

Be sure to make note of the sign change when subtracting the vectors.

$\left \langle -3-(-1),6-2,(-9)-(-9)\right \rangle$

Simplify the vector.

The answer is: $\left \langle -2,4,0\right \rangle$

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Example Question #5 : Vector Subtraction

Find the difference, u-v , of the two vectors.
u=<-3,-7>
v=<6,4>

Possible Answers:

<-9,-11>

<3,-3>

<-3,3>

<9,11>

Correct answer:

<-9,-11>

Explanation:

To find the difference, u-v , we distribute the minus sign and turn it into vector addition.
u-v=<-3,-7>-<6,4>=<-3,-7>+<-6,-4>
The sum of two vectors is defined as

The sum of two vectors

a=<a_1, a_2>

b=<b_1, b_2>

is defined as

a+b=<a_1+b_1, a_2+b_2>

As such, the difference is

<-9,-11>

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Example Question #1 : Vector Subtraction

Compute the following: $\left \langle 1,3,2\right \rangle-\left \langle 2,-4,1\right \rangle$ .

Possible Answers:

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

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

$\left \langle 3,-7,2\right \rangle$

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

Correct answer:

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

Explanation:

The formula for subtracting vectors is $a-b=\left \langle a_1-b_!,a_2-b_2,a_3-b_3\right \rangle$ . Plugging in the values we were given, we get $\left \langle (1-2),(3--4),(2-1)\right \rangle=\left \langle -1,7,1\right \rangle$

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Example Question #2 : Vector Subtraction

Solve:

$\left \langle z, x, y^2z\right \rangle-\left \langle \ln(z), x, 2\right \rangle$

Possible Answers:

$\left \langle z-\ln(z), 1, y^2z-2\right \rangle$

$\left \langle z-\ln(z), 0, y^2z-2\right \rangle$

$z-\ln(z)+y^2z-2$

$\left \langle z, 2x, y^2z-2\right \rangle$

Correct answer:

$\left \langle z-\ln(z), 0, y^2z-2\right \rangle$

Explanation:

To subtract vectors, we simply subtract their respective components:

$\left \langle z-\ln(z), 0, y^2z-2\right \rangle$

Note that our result is a vector still.

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Example Question #293 : Vectors And Vector Operations

Find the difference between $\left \langle 2,0,6\right \rangle$ and $\left \langle 3,3,1\right \rangle$ .

Possible Answers:

$\left \langle 2,0,5\right \rangle$

$\left \langle 0,-1,4\right \rangle$

$\left \langle -1,-3,5\right \rangle$

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

Correct answer:

$\left \langle -1,-3,5\right \rangle$

Explanation:

The formula for the difference of vectors is

$\left \langle x_1,x_2,x_3\right \rangle-\left \langle y_1,y_2,y_3\right \rangle=\left \langle x_1-y_1,x_2-y_2,x_3-y_3\right \rangle$

Using the vectors given, we get

$\left \langle 2-3,0-3,6-1\right \rangle=\left \langle -1,-3,5\right \rangle$ .

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Example Question #3 : Vector Subtraction

Compute the following operation: $\left \langle 2,5,19\right \rangle-\left \langle 3,0,1\right \rangle$ .

Possible Answers:

$\left \langle 17,38,12\right \rangle$

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

$\left \langle -2,6,10\right \rangle$

$\left \langle -1,5,18\right \rangle$

Correct answer:

$\left \langle -1,5,18\right \rangle$

Explanation:

The formula for subtracting vectors a and b is

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

Plugging in the values we were given, we get

$\left \langle (2-3),(5-0),(19-1)\right \rangle=\left \langle -1,5,18\right \rangle$

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Example Question #4 : Vector Subtraction

Perform the following vector operation: $\left \langle 2,12,13\right \rangle-\left \langle 5,5,1\right \rangle$

Possible Answers:

$\left \langle 3,7,12\right \rangle$

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

$\left \langle -3,7,12\right \rangle$

Correct answer:

$\left \langle -3,7,12\right \rangle$

Explanation:

The formula for the difference of vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ is $a-b=\left \langle (x_1-x_2),(y_1-y_2),(z_1-z_2) \right \rangle$ . Using the vectors from the problem statement, we get $\left \langle (2-5),(12-5),(13-1) \right \rangle=\left \langle -3,7,12\right \rangle$

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Example Question #11 : Vector Subtraction

Solve:

$\left \langle xy, z\right \rangle- \left \langle 3xy, z\right \rangle$

Possible Answers:

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

$\left \langle xy, 0\right \rangle$

$\left \langle -2xy, -2z\right \rangle$

-2xy

Correct answer:

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

Explanation:

The difference between two vectors is given by subtracting the corresponding components (for example, $\left \langle a, b\right \rangle - \left \langle x,y\right \rangle=\left \langle a-x, b-y\right \rangle$ )

Our final answer is

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

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #282 : Vectors And Vector Operations

Example Question #3 : Vector Subtraction

Example Question #3 : Vector Subtraction

Example Question #5 : Vector Subtraction

Example Question #1 : Vector Subtraction

Example Question #2 : Vector Subtraction

Example Question #293 : Vectors And Vector Operations

Example Question #3 : Vector Subtraction

Example Question #4 : Vector Subtraction

Example Question #11 : Vector Subtraction

All Calculus 3 Resources

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