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Calculus 3 : Calculus 3

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

Example Questions

Example Question #449 : Vectors And Vector Operations

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

Possible Answers:

x^2-5y+5z^2

x^2+6y+4z^2

x^2-3y+4z^2

x^3+y+7z^2

Correct answer:

x^2-3y+4z^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 use the formula $a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ . Using the vectors in the problem statement, we get (x*x)+(-3*y)+(4z*z)=x^2-3y+4z^2

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Example Question #450 : Vectors And Vector Operations

Consider the vector $\mathbf{v}=\mathbf{i}+7\mathbf{j}+2\mathbf{k}$ .

Which of the following vectors are orthogonal to v?

Possible Answers:

$6\mathbf{i}+-3\mathbf{j}+5\mathbf{k}$

$-\mathbf{i}+3\mathbf{k}$

$4\mathbf{i}-2\mathbf{j}+\mathbf{k}$

$-5\mathbf{i}+2\mathbf{j}+3\mathbf{k}$

$-4\mathbf{i}+3\mathbf{j}-2\mathbf{k}$

Correct answer:

$4\mathbf{i}-2\mathbf{j}+\mathbf{k}$

Explanation:

Two vectors are defined as orthogonal when their dot product is zero.

The dot product of two vectors

$\mathbf{u}=u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k}$ and $\mathbf{v}=v_x\mathbf{i}+v_y\mathbf{j}+v_z\mathbf{k}$

Is given by the expression:

$\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y+u_zv_z$

The only vector that satisfies the requirement that the dot product of it and v is zero is $4\mathbf{i}-2\mathbf{j}+\mathbf{k}$ :

$(3\mathbf{i}+7\mathbf{j}+2\mathbf{k})\cdot (4\mathbf{i}-2\mathbf{j}+\mathbf{k}) =(3)(4)+(7)(-2)+(2)(2)=12-14+2=0$

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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 #2451 : Calculus 3

Solve:

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

Possible Answers:

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

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

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

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

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 #2452 : Calculus 3

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+xyz^2+yz^3

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

3xy+x^2yz^2+yz

3xy+x^2yz^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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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #449 : Vectors And Vector Operations

Example Question #450 : Vectors And Vector Operations

Example Question #451 : Vectors And Vector Operations

Example Question #452 : Vectors And Vector Operations

Example Question #2451 : Calculus 3

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 #2452 : Calculus 3

All Calculus 3 Resources

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