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Calculus 3 : Dot Product

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Example Questions

Example Question #121 : Dot Product

Find the dot product between $\left \langle 5,2,7\right \rangle$ and $\left \langle 2,-1,3\right \rangle$

Possible Answers:

Correct answer:

Explanation:

The formula for the dot product between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ is $a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ . Using the vectors in the problem statement, this becomes (5*2)+(-2*1)+(7*3)=29 .

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Example Question #431 : Vectors And Vector Operations

Solve:

$\left \langle x, x^2+y^2\right \rangle \cdot \left \langle x+1, 5\right \rangle$

Possible Answers:

-4x^2+x-5y^2

6x^2+x+5y^2

5x^2+x+5y^2

2x+6+x^2+y^2

Correct answer:

6x^2+x+5y^2

Explanation:

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

Using this, we get

x^2+x+5x^2+5y^2=6x^2+x+5y^2

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Example Question #122 : Dot Product

Find the dot product between the vectors $\left \langle 5,20,8\right \rangle$ and $\left \langle 3,-1,-9\right \rangle$ .

Possible Answers:

$\left \langle 15,-20,-72\right \rangle$

Correct answer:

Explanation:

The formula for the dot product between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ is $a\cdot b=\left ( x_1*x_2\right )+(y_1*y_2)+(z_1*z_2)$ . We then get $\left ( 5*3\right )+(20*-1)+(8*-9)=(15)+(-20)+(-72)=-77$

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Example Question #432 : Vectors And Vector Operations

Solve:

$\left \langle \sec(z^2), 4, x \right \rangle \cdot \left \langle 5, xy, 3y\right \rangle$

Possible Answers:

$5\sec(z^2)+4x+3xy$

$\left \langle 5\sec(z^2), 7xy \right \rangle$

$5\sec(z^2)+7xy$

$5\sec(z^2)+12xy$

Correct answer:

$5\sec(z^2)+7xy$

Explanation:

The dot product of two vectors is given by the sum of the products of the corresponding components (for example $\left \langle a, b, c\right \rangle \cdot \left \langle e, f, g\right \rangle = ae+bf+cg$ )

Our final answer is

$5\sec(z^2)+7xy$

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Example Question #123 : Dot Product

Find the dot product between the vectors $\left \langle 11,-10,8\right \rangle$ and $\left \langle -1,2,-3\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

he formula for the dot product between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ is $a\cdot b=\left ( x_1*x_2\right )+(y_1*y_2)+(z_1*z_2)$ . We then get $\left ( 11*-1\right )+(-10*2)+(8*-3)=(-11)+(-20)+(-24)=-55$

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Example Question #124 : Dot Product

Find the dot product between the vectors $\left \langle 3,10,5\right \rangle$ and $\left \langle 1,2,6\right \rangle$

Possible Answers:

$\left \langle 3,20,30\right \rangle$

Correct answer:

Explanation:

The formula for the dot product between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ is $a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ . Using the vectors from the problem statement, we get (3*1)+(10*2)+(6*5)=53

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Example Question #441 : Vectors And Vector Operations

Solve:

$\left \langle \sec(z), 14\right \rangle \cdot \left \langle 3, \tan(z)\right \rangle$

Possible Answers:

$3\sec(z)+14\tan(z)$

$42\sec(z)\tan(z)$

$\left \langle 3\sec(z), 14\tan(z) \right \rangle$

$\sec(z)+\tan(z)$

Correct answer:

$3\sec(z)+14\tan(z)$

Explanation:

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

Our final answer is

$3\sec(z)+14\tan(z)$

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Example Question #443 : Vectors And Vector Operations

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

Possible Answers:

$\left \langle 2,-8,5\right \rangle$

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

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Example Question #442 : Vectors And Vector Operations

Find the dot product between $\left \langle \sin(x),y,5\right \rangle$ and $\left \langle \csc(x),2,4\right \rangle$

Possible Answers:

4y+20

3y+22

2y+21

y+5

Correct answer:

2y+21

Explanation:

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Example Question #442 : Vectors And Vector Operations

Find the dot product between $\left \langle x,2y,z\right \rangle$ and $\left \langle 2,2,1\right \rangle$

Possible Answers:

2x-3y-z

x+3y+2z

x+3y-z

2x+4y+z

Correct answer:

2x+4y+z

Explanation:

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Calculus 3 : Dot Product

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #121 : Dot Product

Example Question #431 : Vectors And Vector Operations

Example Question #122 : Dot Product

Example Question #432 : Vectors And Vector Operations

Example Question #123 : Dot Product

Example Question #124 : Dot Product

Example Question #441 : Vectors And Vector Operations

Example Question #443 : Vectors And Vector Operations

Example Question #442 : Vectors And Vector Operations

Example Question #442 : Vectors And Vector Operations

All Calculus 3 Resources

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