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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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:

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

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

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

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

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:

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

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

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

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

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:

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

xz+2x^3y^2

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

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

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:

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

$e^{xy}+2$

$2e^{xy}$

$2e^{xy}+1$

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:

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

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

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

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

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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Example Question #426 : Vectors And Vector Operations

Solve:

$\left \langle 3\cos^2(x), 5e^y\right \rangle \cdot \left \langle 4\sin^2(x), 1\right \rangle$

Possible Answers:

$12\cos^2(x)\sin^2(x)+e^y$

$12\cos^2(x)\sin^2(x)+5e^y$

12+5e^y

$\left \langle12\cos^2(x)\sin^2(x), 5e^y \right \rangle$

Correct answer:

$12\cos^2(x)\sin^2(x)+5e^y$

Explanation:

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

Our answer is

$12\cos^2(x)\sin^2(x)+5e^y$

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Example Question #427 : Vectors And Vector Operations

Find the dot product.

u=<2,7>

v=<5,10>

Possible Answers:

240

700

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

$u\bullet v=2(5)+7(10)=10+70=80$

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Example Question #428 : Vectors And Vector Operations

Find the dot product.

u=<1,0>

v=<3,2>

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

$u\bullet v=1(3)+0(2)=3+0=3$

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Example Question #421 : Vectors And Vector Operations

Compute $a\cdot b$ , where $a=\left \langle 3,0,4\right \rangle$ and $b=\left \langle 9,11,2\right \rangle$ .

Possible Answers:

$\left \langle 2,11,2\right \rangle$

$\left \langle -6,-11,4\right \rangle$

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

(3*9)+(0*11)+(4*2)=35 .

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Example Question #2432 : Calculus 3

Compute the dot product of the vectors $\left \langle 2,9,21\right \rangle$ and $\left \langle 2,5,0\right \rangle$ .

Possible Answers:

$\left \langle 4,45,0\right \rangle$

Correct answer:

Explanation:

The formula for the dot product of vectors

$a=\left \langle x_1,y_1,z_1\right \rangle$

and

$b=\left \langle x_2,y_2,z_2\right \rangle$

$a\cdot b=(x_1*x_2)+(y_1*y_2)+(z_1*z_2)$ .

Using the vectors we were given, this becomes

(2*2)+(9*5)+(21*0) which equals

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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 #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

Example Question #426 : Vectors And Vector Operations

Example Question #427 : Vectors And Vector Operations

Example Question #428 : Vectors And Vector Operations

Example Question #421 : Vectors And Vector Operations

Example Question #2432 : Calculus 3

All Calculus 3 Resources

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