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Calculus 3 : Vectors and Vector Operations

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

Example Questions

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 #291 : Vectors And Vector Operations

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

Possible Answers:

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

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

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

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

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 #292 : Vectors And Vector Operations

Solve:

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

Possible Answers:

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

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

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

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

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 #293 : Vectors And Vector Operations

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

Possible Answers:

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

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

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

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

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 #294 : Vectors And Vector Operations

Solve:

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

Possible Answers:

-2xy

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

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

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

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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Example Question #2301 : Calculus 3

Find the difference between the vectors $\left \langle 4x,7y,-4z\right \rangle$ and $\left \langle 2x,0,-z\right \rangle$

Possible Answers:

$\left \langle -x,7y,-3z\right \rangle$

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

$\left \langle 2x,7y,-5z\right \rangle$

$\left \langle 2x,y,-3z\right \rangle$

Correct answer:

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

Explanation:

To find the difference between the vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the following formula:

$a-b=\left \langle x_1-x_2,y_1-y_2,z_1-z_2\right \rangle$

Using the vectors from the problem statement, we get

$\left \langle 4x-2x,7y-0,-4z+z\right \rangle=\left \langle2x,7y,-3z\right \rangle$

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Example Question #13 : Vector Subtraction

Find the difference between the vectors $\left \langle -10,-20,30\right \rangle$ and $\left \langle 27,15,60\right \rangle$

Possible Answers:

$\left \langle -37,-35,-30\right \rangle$

$\left \langle -37,-20,-30\right \rangle$

$\left \langle -47,-35,-30\right \rangle$

$\left \langle -37,-32,-20\right \rangle$

Correct answer:

$\left \langle -37,-35,-30\right \rangle$

Explanation:

To find the difference between the vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the following formula:

$a-b=\left \langle x_1-x_2,y_1-y_2,z_1-z_2\right \rangle$

Using the vectors from the problem statement, we get

$\left \langle-10-27,-20-15,30-60\right \rangle=\left \langle-37,-35,-30\right \rangle$

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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 #3 : Vector Subtraction

Example Question #5 : Vector Subtraction

Example Question #291 : Vectors And Vector Operations

Example Question #292 : Vectors And Vector Operations

Example Question #293 : Vectors And Vector Operations

Example Question #293 : Vectors And Vector Operations

Example Question #4 : Vector Subtraction

Example Question #294 : Vectors And Vector Operations

Example Question #2301 : Calculus 3

Example Question #13 : Vector Subtraction

All Calculus 3 Resources

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