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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #451 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle 3x,y,5z\right \rangle$ and $\left \langle x^2,y^3,z^4\right \rangle$

Possible Answers:

3x^2+y^3+z^4

x^3+y+z

3x^3+y^4+5z^5

3x+y+5z+x^2+y^3+z^4

Correct answer:

3x^3+y^4+5z^5

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 get

(3x*x^2)+(y*y^3)+(5z*z^4)=3x^3+y^4+5z^5

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Example Question #452 : Vectors And Vector Operations

Find the dot product between the vectors $\left \langle -4,-10,1\right \rangle$ and $\left \langle x,y^2,z\right \rangle$

Possible Answers:

-4x-10y^2+z

4x^2-10y^2+1

4-10y+z

4x^3-5y^2+z

Correct answer:

-4x-10y^2+z

Explanation:

o 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 get

(-4*x)+(-10*y^2)+(1*z)=-4x-10y^2+z

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Example Question #451 : Vectors And Vector Operations

Solve:

$\left \langle z, y^3z^2, x\right \rangle \cdot \left \langle 2, e^x, x\right \rangle$

Possible Answers:

2z+e^xy^3z^2+x^2

2z+e^xy^3z^2+2x

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

2z+e^xy^3z^2+x

Correct answer:

2z+e^xy^3z^2+x^2

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

2z+e^xy^3z^2+x^2

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Example Question #454 : Vectors And Vector Operations

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

Possible Answers:

3x-7y+4z

3x-7y-z

2x-7y+4z

3x-y+4z

Correct answer:

3x-7y+4z

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

(x*3)+(y*-7)+(4z*1)=3x-7y+4z

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Example Question #132 : Dot Product

Find the dot product between the vectors $\left \langle 4,1,-5\right \rangle$ and $\left \langle 2,-8,1\right \rangle$

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

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

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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 #452 : Vectors And Vector Operations

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:

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

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

3xy+xyz^2+yz^3

3xy+x^2yz^2+yz

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 #453 : Vectors And Vector Operations

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 #454 : Vectors And Vector Operations

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

Possible Answers:

573

None of the other answers

5958

200

$\approx 148.781$

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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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #451 : Vectors And Vector Operations

Example Question #452 : Vectors And Vector Operations

Example Question #451 : Vectors And Vector Operations

Example Question #454 : Vectors And Vector Operations

Example Question #132 : Dot Product

Example Question #452 : Vectors And Vector Operations

Example Question #453 : Vectors And Vector Operations

Example Question #452 : Vectors And Vector Operations

Example Question #453 : Vectors And Vector Operations

Example Question #454 : Vectors And Vector Operations

All Calculus 3 Resources

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