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Calculus 3 : Dot Product

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Example Questions

Example Question #452 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle 3,0,-6\right \rangle$ and $\left \langle 1,-2,3\right \rangle$

Possible Answers:

Correct answer:

Explanation:

To find the dot product between vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , you apply the following formula:

$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ . Applying to the vectors from the problem statement, we get:

(3*1)+(0*-2)+(-6*3)=-15

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Example Question #453 : Vectors And Vector Operations

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

Possible Answers:

4x^6-19

7x-10

x^4-16

3x^6-17

Correct answer:

4x^6-19

Explanation:

To find the dot product between vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , you apply the following formula:

$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ . Applying to the vectors from the problem statement, we get:

(x^3*4x^3)+(3*-7)+(1*2)=4x^6-19

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Example Question #141 : Dot Product

Find the dot product of the two vectors:

$\left \langle 3, xz, yz^2\right \rangle \cdot \left \langle xy, xyz, z\right \rangle$

Possible Answers:

$\left \langle 3xy, x^2yz^2,yz^3 \right \rangle$

3xy+x^2yz^2+yz^3

3xy+x^2yz^2+yz

3xy+xyz^2+yz^3

Correct answer:

3xy+x^2yz^2+yz^3

Explanation:

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

Our final answer is

3xy+x^2yz^2+yz^3

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Example Question #142 : Dot Product

What is the product of $\vec {a}=<-2,0,7>$ and $\vec {b}=<-3,4,1>$

Possible Answers:

Correct answer:

Explanation:

Screen shot 2016 01 07 at 8.41.42 pm

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Example Question #143 : Dot Product

Evaluate the dot product of the vectors <23, 78>,<42,64> .

Possible Answers:

None of the other answers

$\approx 148.781$

5958

200

573

Correct answer:

5958

Explanation:

To evaluate the dot product of any two vectors, we multiply them component-wise.

$23\times42 +78 \times 64 = 5958$ .

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Example Question #144 : Dot Product

Solve:

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

Possible Answers:

$\left \langle 4\right \rangle$

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

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 #145 : Dot Product

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 #146 : Dot Product

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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Calculus 3 : Dot Product

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #452 : Vectors And Vector Operations

Example Question #453 : Vectors And Vector Operations

Example Question #141 : Dot Product

Example Question #142 : Dot Product

Example Question #143 : Dot Product

Example Question #144 : Dot Product

Example Question #145 : Dot Product

Example Question #463 : Vectors And Vector Operations

Example Question #146 : Dot Product

Example Question #465 : Vectors And Vector Operations

All Calculus 3 Resources

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