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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #2471 : Calculus 3

Find the dot product of the two vectors.

u=<-5,4>

v=<-3,-3>

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors

a=<a_1,a_2>

b=<b_1,b_2>

is defined as

$a \bullet b=a_1*b_1+a_2*b_2$

For the vectors in this problem we have

$u \bullet v=-5*(-3)+4*(-3)=15-12=3$

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Example Question #2472 : Calculus 3

Find the dot product of the two vectors.

u=<6,-7>

v=<-4,-1>

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors

a=<a_1,a_2>

b=<b_1,b_2>

is defined as

$a \bullet b=a_1*b_1+a_2*b_2$

For the vectors in this problem we have

$u \bullet v=6*(-4)+(-7)*(-1)=-24+7=-17$

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Example Question #2473 : Calculus 3

Find the dot product between the vectors $\left \langle 2,y,7\right \rangle$ and $\left \langle 3,4,-1\right \rangle$

Possible Answers:

4y-1

4y-5

y-1

3y-2

Correct answer:

4y-1

Explanation:

To find the dot product between two 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 apply the following formula:

$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we get

(2*3)+(y*4)+(7*-1)=4y-1

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Example Question #2474 : Calculus 3

Find the dot product between the vectors $\left \langle 5,4y,6\right \rangle$ and $\left \langle 4,4,-3\right \rangle$

Possible Answers:

16y+3

8y+2

16y+2

16y+5

Correct answer:

16y+2

Explanation:

To find the dot product between two 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 apply the following formula:

$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we get

(5*4)+(4y*4)+(6*-3)=16y+2

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Example Question #2471 : Calculus 3

Find the dot product between the vectors $\left \langle 3,5,1\right \rangle$ and $\left \langle 6,2,1\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product between two 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 apply the formula

$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we get

(3*6)+(5*2)+(1*1)=29

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Example Question #2475 : Calculus 3

Find the dot product between the vectors $\left \langle 2,0,8\right \rangle$ and $\left \langle 1,-4,2\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product between two 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 apply the formula

$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we get

(2*1)+(0*-4)+(8*2)=18

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Example Question #471 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle -7,-1,0\right \rangle$ and $\left \langle 2,3,0\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product between two 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 apply the following formula:
$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$

Using the vectors from the problem statement, we get

(-7*2)+(-1*3)+(0*0)=-17

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Example Question #472 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle -8,-2,1\right \rangle$ and $\left \langle 5,3,1 \right \rangle$

Possible Answers:

Correct answer:

Explanation:

Using the vectors from the problem statement, we get

(-8*5)+(-2*3)+(1*1)=-45

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Example Question #471 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle 2,14,1\right \rangle$ and $\left \langle -5,0,9\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product between the 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:

$a\cdot b=(a_1b_1)+(a_2b_2)+(a_3b_3)$

Using the vectors from the problem statement and applying, we get

(2*-5)+(14*0)+(1*9)=-1

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Example Question #472 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle 2,5,-3\right \rangle$ and $\left \langle 4,-1,1\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product between the 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:

$a\cdot b=(a_1b_1)+(a_2b_2)+(a_3b_3)$

Using the vectors from the problem statement and applying, we get

(2*4)+(5*-1)+(-3*1)=0

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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 #2471 : Calculus 3

Example Question #2472 : Calculus 3

Example Question #2473 : Calculus 3

Example Question #2474 : Calculus 3

Example Question #2471 : Calculus 3

Example Question #2475 : Calculus 3

Example Question #471 : Vectors And Vector Operations

Example Question #472 : Vectors And Vector Operations

Example Question #471 : Vectors And Vector Operations

Example Question #472 : Vectors And Vector Operations

All Calculus 3 Resources

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