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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #241 : Vectors And Vector Operations

Find the sum, u+v

u=<4,16>

v=<2,4>

Possible Answers:

<6,20>

<8,64>

<2,12>

Correct answer:

<6,20>

Explanation:

The sum of two vectors

a=<a_1, a_2>

b=<b_1, b_2>

is defined as

a+b=<a_1+b_1, a_2+b_2>

For the vectors in this problem

u+v=<4+2, 16+4>

=<6,20>

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Example Question #35 : Vector Addition

Perform the matrix operation $\begin{bmatrix} 1&3 \\ 4&0 \end{bmatrix}+\begin{bmatrix} 3&9 \\ 17&38 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 4&12 \\ 21 &38 \end{bmatrix}$

$\begin{bmatrix} 19&17 \\ 11& 10 \end{bmatrix}$

$\begin{bmatrix} -4&11 \\ 9&1 \end{bmatrix}$

$\begin{bmatrix} 5&3 \\ 17&38 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 4&12 \\ 21 &38 \end{bmatrix}$

Explanation:

To find the sum $\begin{bmatrix} a&b \\ c&d \end{bmatrix}+\begin{bmatrix} e& f\\ g &h \end{bmatrix}=\begin{bmatrix} a+e& b+f\\ c+g&d+h \end{bmatrix}$ . Using the matrices in the problem statement, we get $\begin{bmatrix} 1+3& 3+9\\ 4+17 &0+38 \end{bmatrix}$ = $\begin{bmatrix} 4&12 \\ 21 &38 \end{bmatrix}$

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Example Question #241 : Vectors And Vector Operations

Solve:

$\left \langle 10, x\sin^2(y)\right \rangle+ \left \langle x, x\cos^2(y)\right \rangle$

Possible Answers:

$\left \langle 10+x, x\right \rangle$

10+2x

$10x+x^2\sin^2(y)\cos^2(y)$

$\left \langle 10x, x^2 \right \rangle$

Correct answer:

$\left \langle 10+x, x\right \rangle$

Explanation:

When adding vectors, we simply add the corresponding components together:

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

Our answer is

$\left \langle 10+x, x\right \rangle$

Note that our final answer is a vector.

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Example Question #37 : Vector Addition

Solve:

$\left \langle z^2e^z, \cos(x)\right \rangle+ \left \langle ze^z, \sin(x)\right \rangle$

Possible Answers:

$e^z(z^2+z)+ \cos(x)+\sin(x)$

$\left \langle e^z(z^2+z),1\right \rangle$

$\left \langle e^{2z}z^2z, \cos(x)\sin(x)\right \rangle$

$\left \langle e^z(z^2+z), \cos(x)+\sin(x)\right \rangle$

Correct answer:

$\left \langle e^z(z^2+z), \cos(x)+\sin(x)\right \rangle$

Explanation:

The sum of two vectors is the vector given by the sums of the respective components (for example, $\left \langle a, b\right \rangle+\left \langle c, d\right \rangle=\left \langle a+c, b+d\right \rangle$ ).

Our final answer is

$\left \langle e^z(z^2+z), \cos(x)+\sin(x)\right \rangle$

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Example Question #38 : Vector Addition

Perform the following vector operation: $\left \langle 2xy,3y,z^3\right \rangle+\left \langle 7xy,9y,2z^3\right \rangle$

Possible Answers:

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

$\left \langle 14xy,27y,2z^3\right \rangle$

$\left \langle 8xy,10y,3z^3\right \rangle$

$\left \langle 9xy,12y,3z^3\right \rangle$

Correct answer:

$\left \langle 9xy,12y,3z^3\right \rangle$

Explanation:

The formula for adding 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 $a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle$ . Using the vectors from the problem statement, this becomes $\left \langle 2xy+7xy,3y+9y,z^3+2z^3\right \rangle=\left \langle 9xy,12y,3z^3\right \rangle$

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Example Question #39 : Vector Addition

Solve:

$\left \langle \cos(yz), e^{z}\right \rangle + \left \langle 3\cos(yz), 2e^x\right \rangle$

Possible Answers:

$\left \langle 4\cos(yz), 3e^z\right \rangle$

$4\cos(yz)+e^z+2e^x$

$\left \langle 16\cos(yz)^2, e^{2z}+4e^{2x}\right \rangle$

$\left \langle 4\cos(yz), e^z+2e^x\right \rangle$

Correct answer:

$\left \langle 4\cos(yz), e^z+2e^x\right \rangle$

Explanation:

To add two vectors, we simply add the corresponding components (for example, $\left \langle a, b\right \rangle+ \left \langle c, d\right \rangle=\left \langle a+c, b+d\right \rangle$ )

Our final answer is

$\left \langle 4\cos(yz), e^z+2e^x\right \rangle$

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Example Question #40 : Vector Addition

Find the sum of the vectors $\left \langle 3,y^2,\sin(z)\right \rangle$ and $\left \langle 5,5y^2,3\sin(z)\right \rangle$

Possible Answers:

$\left \langle 6,6y^2,4\cos(z)\right \rangle$

$\left \langle 8,3y^2,4\sin(z)\right \rangle\,$

$\left \langle 7,6y^2,\sin(z)\right \rangle\textup{}$

$\left \langle 8,6y^2,4\sin(z)\right \rangle$

Correct answer:

$\left \langle 8,6y^2,4\sin(z)\right \rangle$

Explanation:

To find the sum of 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 $a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle$ . Using the vectors we were given, we get $a+b=\left \langle 3+5,y^2+5y^2,\sin(z)+3\sin(z)\right \rangle=\left \langle 8,6y^2,4\sin(z)\right \rangle$

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Example Question #41 : Vector Addition

Find the sum of the vectors $\left \langle 2,y^4,xyz)\right \rangle$ and $\left \langle 10,3y^4,2xyz\right \rangle$

Possible Answers:

$\left \langle 12,y^4,3xyz\right \rangle$

$\left \langle 11,4y^3,3xyz\right \rangle$

$\left \langle 12,4y^4,3xyz\right \rangle$

$\left \langle 12,4y^4,2xyz\right \rangle$

Correct answer:

$\left \langle 12,4y^4,3xyz\right \rangle$

Explanation:

To find the sum of 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 $a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle$ . Using the vectors we were given, we get $a+b=\left \langle 2+10,y^4+3y^4,xyz+2xyz\right \rangle=\left \langle 12,4y^4,3xyz\right \rangle$

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Example Question #243 : Vectors And Vector Operations

Find the sum of the vectors $\left \langle -4,3y^3,z\right \rangle$ and $\left \langle 2,y^3,5z\right \rangle$

Possible Answers:

-2+4y^3+6z

$\left \langle -2,2y^3,3z\right \rangle$

$\left \langle -2,4y^3,6z\right \rangle$

$\left \langle -8,3y^6,5z^2\right \rangle$

Correct answer:

$\left \langle -2,4y^3,6z\right \rangle$

Explanation:

To find the sum of 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 $a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle$ . Using the vectors we were given, we get $a+b=\left \langle -4+2,3y^3+y^3,z+5z\right \rangle=\left \langle -2,4y^3,6z\right \rangle$

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Example Question #41 : Vector Addition

Find the sum of the vectors $\left \langle 5,3xyz,1\right \rangle$ and $\left \langle -4,2xyz,-5\right \rangle$

Possible Answers:

1+5xyz+4

$\left \langle 2,3xyz,4\right \rangle$

$\left \langle 1,5xyz,-4\right \rangle$

$\left \langle 0,xyz,1\right \rangle$

Correct answer:

$\left \langle 1,5xyz,-4\right \rangle$

Explanation:

To find the sum of 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 $a+b=\left \langle x_1+x_2,y_1+y_2,z_1+z_2\right \rangle$ . Using the vectors we were given, we get $a+b=\left \langle 5-4,3xyz+2xyz,1-5\right \rangle=\left \langle 1,5xyz,-4\right \rangle$

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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 #241 : Vectors And Vector Operations

Example Question #35 : Vector Addition

Example Question #241 : Vectors And Vector Operations

Example Question #37 : Vector Addition

Example Question #38 : Vector Addition

Example Question #39 : Vector Addition

Example Question #40 : Vector Addition

Example Question #41 : Vector Addition

Example Question #243 : Vectors And Vector Operations

Example Question #41 : Vector Addition

All Calculus 3 Resources

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