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Calculus 3 : Dot Product

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #413 : Vectors And Vector Operations

Solve:

$\left \langle 0, 1, 1 \right \rangle \cdot \left \langle 3, 2, 5 \right \rangle$

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is given by

$\left \langle a, b, c \right \rangle \cdot \left \langle d, e, f \right \rangle= ad+be+cf$

Using this, we get

0+2+5=7

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Example Question #414 : Vectors And Vector Operations

Solve:

$\left \langle 1, 1, 2 \right \rangle \cdot \left \langle \cos(2), 0, 2 \right \rangle$

Possible Answers:

$4\cos(2)$

$\cos(2)+5$

$\cos(2)+4$

Correct answer:

$\cos(2)+4$

Explanation:

The dot product of two vectors is given by

$\left \langle a, b,c \right \rangle \cdot \left \langle d, e, f \right \rangle = ad+be+cf$

Using this, we get

$\cos(2)+0+4=\cos(2)+4$

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Example Question #415 : Vectors And Vector Operations

Solve:

$\left \langle x, 2xy \right \rangle\cdot \left \langle ze^{y}, 3x^2 \right \rangle$

Possible Answers:

6x^4yze^y

xze^y+6x^2y

xze^y-6x^3y

$xze^{y}+6x^3y$

Correct answer:

$xze^{y}+6x^3y$

Explanation:

The dot product of two vectors is given by

$\left \langle a, b \right \rangle \cdot \left \langle c, d \right \rangle= ac+bd$

Our final answer is

$xze^{y}+6x^3y$

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Example Question #416 : Vectors And Vector Operations

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

Possible Answers:

Correct answer:

Explanation:

The formula for the dot product is $a\cdot b=(i_1*i_2)+(j_1*j_2)+(k_1*k_2)$ . Using the given vectors, we get (2*1)+(-5*1)+(1*3)=0 .

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Example Question #2422 : Calculus 3

Compute the dot product between $\left \langle 1,2,3\right \rangle$ and $\left \langle 3,0,1\right \rangle$ .

Possible Answers:

$\left \langle 4,2,4\right \rangle$

$\left \langle -1,2,4\right \rangle$

$\left \langle 3,0,3\right \rangle$

$\left \langle 0,2,3\right \rangle$

Correct answer:

$\left \langle 3,0,3\right \rangle$

Explanation:

The formula for the dot product is $a\cdot b=(i_1*i_2)+(j_1*j_2)+(k_1*k_2)$ . Using the two given vectors and solving, $a\cdot b=\left \langle 1*3,2*0,3*1\right \rangle=\left \langle 3,0,3\right \rangle$ .

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Example Question #421 : Vectors And Vector Operations

Solve:

$\left \langle xz, z^3\right \rangle\cdot \left \langle 2xy, \tan(x)\right \rangle$

Possible Answers:

$\frac{2x^2yz}{z^3\tan(x)}$

$2x^2yz^4\tan(x)$

$2x^2yz-z^3\tan(x)$

$2x^2yz+z^3\tan(x)$

Correct answer:

$2x^2yz+z^3\tan(x)$

Explanation:

The dot product of two vectors is given by

$\left \langle a, b\right \rangle \cdot \left \langle c, d\right \rangle=ac+bd$

So, our answer is

$2x^2yz+z^3\tan(x)$

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Example Question #422 : Vectors And Vector Operations

Solve:

$\left \langle \cos(x), xyz\right \rangle \cdot \left \langle x^2, y\right \rangle$

Possible Answers:

$\left \langle x^2\cos(x), xy^2z\right \rangle$

$x^2\cos(x)+xyz$

$x^2\cos(x)+xy^2z$

$\cos(x^3)+xy^2z$

Correct answer:

$x^2\cos(x)+xy^2z$

Explanation:

The dot product of two vectors is found by adding the product of the corresponding components. (For example, $\left \langle a, b\right \rangle \cdot \left \langle c, d\right \rangle=ac+bd$ .)

For our vectors, we get

$x^2\cos(x)+xy^2z$

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Example Question #423 : Vectors And Vector Operations

Solve:

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

Possible Answers:

xz+x^3y^2+x^2y^3

$\left \langle zx, x^3y^2 + x^2y^3\right \rangle$

$\left \langle zx, x^3y^2,x^2y^3\right \rangle$

xz+2x^3y^2

Correct answer:

xz+x^3y^2+x^2y^3

Explanation:

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

So, we get

xz+x^3y^2+x^2y^3

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Example Question #424 : Vectors And Vector Operations

Solve:

$\left \langle e^{xy}, \sin^2(y), 1\right \rangle \cdot \left \langle 2, 1, \cos^2(y)\right \rangle$

Possible Answers:

$e^{xy}+2$

$2e^{xy}$

$e^{xy}+2\sin^2(y)+\cos^2(y)$

$2e^{xy}+1$

Correct answer:

$2e^{xy}+1$

Explanation:

The dot product of two vectors is given by

$\left \langle a, b, c\right \rangle \cdot \left \langle d, e,f \right \rangle=ad+be+cf$

So, we get

$2e^{xy}+1$

We used a Pythagorean trigonometric identity to simplify.

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Example Question #425 : Vectors And Vector Operations

Solve:

$\left \langle e^{xy}, \sin(xz)\right \rangle \cdot \left \langle e^{2x}, 3\right \rangle$

Possible Answers:

$\left \langle e^{2x+xy}, 3\sin(xz) \right \rangle$

$e^{2x^2y}+3\sin(xz)$

$e^{2x+xy}+\sin(xz)$

$e^{2x+xy}+3\sin(xz)$

Correct answer:

$e^{2x+xy}+3\sin(xz)$

Explanation:

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

Our final answer is therefore

$e^{2x+xy}+3\sin(xz)$

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Calculus 3 : Dot Product

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #413 : Vectors And Vector Operations

Example Question #414 : Vectors And Vector Operations

Example Question #415 : Vectors And Vector Operations

Example Question #416 : Vectors And Vector Operations

Example Question #2422 : Calculus 3

Example Question #421 : Vectors And Vector Operations

Example Question #422 : Vectors And Vector Operations

Example Question #423 : Vectors And Vector Operations

Example Question #424 : Vectors And Vector Operations

Example Question #425 : Vectors And Vector Operations

All Calculus 3 Resources

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