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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #461 : Vectors And Vector Operations

Solve:

$\left \langle \cos^2(z), 4\right \rangle\cdot \left \langle 4, \sin^2(z)\right \rangle$

Possible Answers:

$\left \langle 4\cos^2(z), 4\sin^2(z)\right \rangle$

$\left \langle 4\right \rangle$

Correct answer:

Explanation:

The dot product of two vectors is given by the sum of the products of their respective components (for example, $\left \langle a, b\right \rangle \cdot \left \langle c,d \right \rangle = ac+bd$ )

Our final answer is

$4(\cos^2(z)+\sin^2(z))=4$

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Example Question #462 : Vectors And Vector Operations

Evaluate the dot product of the two vectors.

u=<1,4>

v=<-2,3>

Possible Answers:

Correct answer:

Explanation:

The dot product for the 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 find that

$u\bullet v=1*(-2)+4(3)=-2+12=10$

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Example Question #463 : Vectors And Vector Operations

Evaluate the dot product of the two vectors

u=<-2,-7>

v=<3,5>

Possible Answers:

210

Correct answer:

Explanation:

The dot product for the 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 find that

$u\bullet v=-2*3+(-7)*5=-6-35=-41$

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Example Question #464 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle 5,2,1\right \rangle$ and $\left \langle 4,-3,-6\right \rangle$

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 $\left \langle x_2,y_2,z_2\right \rangle$ , we use 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

(5*4)+(2*-3)+(1*-6)=8

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Example Question #465 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle 15,12,10\right \rangle$ and $\left \langle 1,3,2\right \rangle$

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 $\left \langle x_2,y_2,z_2\right \rangle$ , we use 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

(15*1)+(12*3)+(10*2)=71

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Example Question #466 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle 2,7,20\right \rangle$ and $\left \langle 5,6,1\right \rangle$

Possible Answers:

Correct answer:

Explanation:

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

Using the vectors from the problem statement, we then get

(2*5)+(7*6)+(20*1)=72

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Example Question #2464 : Calculus 3

Find the dot product between the vectors $\left \langle 1,9,6\right \rangle$ and $\left \langle -3,10,1\right \rangle$

Possible Answers:

Correct answer:

Explanation:

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

Using the vectors from the problem statement, we then get

(1*-3)+(9*10)+(6*1)=93

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Example Question #467 : Vectors And Vector Operations

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

Possible Answers:

$\left \langle 6,y^4,5\right \rangle$

11+y^4

12+3y^2

11-y^4

Correct answer:

11+y^4

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

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

(3*2)+(y*y^3)+(1*5)=11+y^4

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Example Question #462 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle 9,9,3\right \rangle$ and $\left \langle x,2,1\right \rangle$

Possible Answers:

7x+21

9x+18

10x+21

9x+21

Correct answer:

9x+21

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

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

(9*x)+(9*2)+(3*1)=9x+21

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Example Question #462 : Vectors And Vector Operations

Find the dot product of the two vectors.

u=<10,-8>

v=<3,-1>

Possible Answers:

240

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

$u \bullet v=10*3+(-8)*(-1)=30+8=38$

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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 #461 : Vectors And Vector Operations

Example Question #462 : Vectors And Vector Operations

Example Question #463 : Vectors And Vector Operations

Example Question #464 : Vectors And Vector Operations

Example Question #465 : Vectors And Vector Operations

Example Question #466 : Vectors And Vector Operations

Example Question #2464 : Calculus 3

Example Question #467 : Vectors And Vector Operations

Example Question #462 : Vectors And Vector Operations

Example Question #462 : Vectors And Vector Operations

All Calculus 3 Resources

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