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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

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

Example Questions

Example Question #721 : Vectors And Vector Operations

Find the determinant of the 2x2 matrix $\begin{bmatrix} 4& 5\\ 1& 20 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

The formula for the determinant of a 2x2 matrix $a=\begin{bmatrix} a&b \\ c& d \end{bmatrix}$ is $\left | a\right |=ad-cb$ . Using the matrix from the problem statement, we get (20*4)-(1*5)=75

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Example Question #722 : Vectors And Vector Operations

Find the determinant of the 2x2 matrix $\begin{bmatrix} 1&14 \\ 2& 5 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

The formula foe the determinant of a 2x2 matrix $\begin{bmatrix} a&b \\ c& d \end{bmatrix}$ is ad-cb . Using the matrix from the problem statement, we get (1*5)-(2*14)=5-28=-23

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Example Question #723 : Vectors And Vector Operations

Find the determinant of the given matrix:

$\begin{vmatrix} x& y\\ \sin(z)&\cos(x) \end{vmatrix}$

Possible Answers:

$x\cos(x)+y\sin(z)$

$x\cos(x)-y\sin(z)$

$\left \langle x\cos(x), -y\sin(z) \right \rangle$

Correct answer:

$x\cos(x)-y\sin(z)$

Explanation:

The determinant of the matrix is found by multiplying from top left to bottom right, and then subtracting the product from top right to bottom left (for example, the determinant of $\begin{vmatrix} a&b \\ c& d \end{vmatrix}=ad-bc$ )

Our answer is

$x\cos(x)-y\sin(z)$

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Example Question #724 : Vectors And Vector Operations

Find the determinant of the 2x2 matrix $\begin{bmatrix} 4& x\\ 2y& z \end{bmatrix}$

Possible Answers:

2x-3y

4-2xy

4z-2yx

xy-4xyz

Correct answer:

4z-2yx

Explanation:

To find the determinant of a 2x2 matrix $a=\begin{bmatrix} a&b \\ c&d \end{bmatrix}$ we use the formula for the determinant, which is (a*d)-(c*b) . Plugging in the values from the problem statement, we find that the determinant of the matrix is (4*z)-(2y*x)=4x-2yx

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Example Question #725 : Vectors And Vector Operations

Find the determinant of the 3x3 matrix $\begin{bmatrix} 1& 3& -1\\ 5&0 &3 \\ -3&2 &7 \end{bmatrix}$

Possible Answers:

120

112

148

Correct answer:

Explanation:

To find the determinant of any 3x3 matrix $a=\begin{bmatrix} a&b &c \\ d& e& f\\ g& h& i \end{bmatrix}$ , we use the formula determinant=a(ei-hf)-b(di-gf)+c(dh-ge) . Using the vector from the problem statement, we get 1(0-6)-3(35+9)-1(10-0)=-6-132-10=-148

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Example Question #726 : Vectors And Vector Operations

Perform the matrix operation $\begin{bmatrix} 2&1 \\ 3& 6 \end{bmatrix}-\begin{bmatrix} 0&7 \\ 1&5 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 2&-6 \\ 2 & 1 \end{bmatrix}$

$\begin{bmatrix} 2&5 \\ 2 & 1 \end{bmatrix}$

$\begin{bmatrix} 1&-6 \\ 0& 1 \end{bmatrix}$

$\begin{bmatrix} 2&-6 \\ 2 & 4 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 2&-6 \\ 2 & 1 \end{bmatrix}$

Explanation:

To find the difference between two 2x2 matrices $a=\begin{bmatrix} a& b\\ c&d \end{bmatrix}$ and $b=\begin{bmatrix} e& f\\ g& h \end{bmatrix}$ you use the formula $a-b=\begin{bmatrix} a-e&b-f \\ c-g& d-h \end{bmatrix}$ . Using the two matrices from the problem statement, we then get $\begin{bmatrix} 2-0& 1-7\\ 3-1& 6-5 \end{bmatrix}=\begin{bmatrix} 2&-6 \\ 2& 1 \end{bmatrix}$

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Example Question #727 : Vectors And Vector Operations

Perform the matrix operation $\begin{bmatrix} 4&x \\ z& 5 \end{bmatrix}-\begin{bmatrix} 1&5x \\ 3z&7 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 7&3x \\ -2z&0 \end{bmatrix}$

$\begin{bmatrix} 3&-4x \\ -2z&-2 \end{bmatrix}$

$\begin{bmatrix} 1&x \\ -3z&5 \end{bmatrix}$

$\begin{bmatrix} -4&3x \\ z&6 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 3&-4x \\ -2z&-2 \end{bmatrix}$

Explanation:

To find the difference between two 2x2 matrices $a=\begin{bmatrix} a& b\\ c&d \end{bmatrix}$ and $b=\begin{bmatrix} e& f\\ g& h \end{bmatrix}$ you use the formula $a-b=\begin{bmatrix} a-e&b-f \\ c-g& d-h \end{bmatrix}$ . Using the two matrices from the problem statement, we then get $\begin{bmatrix} 4-1& x-5x\\ z-3z& 5-7 \end{bmatrix}=\begin{bmatrix} 3&-4x \\ -2z& -2\end{bmatrix}$

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Example Question #726 : Vectors And Vector Operations

Find the determinant of the 2x2 matrix $\begin{bmatrix} 4& 2\\ 1&-5 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

To find the determinant of a 2x2 matrix $a=\begin{bmatrix} a&b \\ c&d \end{bmatrix}$ ,we use the formula

determinant=(ad)-(cb)

Using hte matrix from the problem statement, we get

determinant=(-4*5)-(1*2)=-22

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Example Question #729 : Vectors And Vector Operations

Find the determinant of the 3x3 matrix $\begin{bmatrix} i&j &k \\ 3&-4 &1 \\ 2&0 &5 \end{bmatrix}$

Possible Answers:

$\left \langle 20,15,8\right \rangle$

$\left \langle -20,-13,8\right \rangle$

$\left \langle -20,13,8\right \rangle$

$\left \langle 4,-10,8\right \rangle$

Correct answer:

$\left \langle -20,-13,8\right \rangle$

Explanation:

To find the determinant of the 3x3 matrix $\begin{bmatrix} i& j& k\\ a&b &c \\ x&y &z \end{bmatrix}$ , we use the formula:

determinant=i(bz-yc)-j(az-xc)+k(ay-xb) . Using the matrix from the problem statement, we get:

$i(-20-0)-j(15-2)+k(0+8)=\left \langle -20,-13,8\right \rangle$

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Example Question #730 : Vectors And Vector Operations

Find the determinant of the 3x3 matrix $\begin{bmatrix} i&j &k \\ 2&1 &1 \\ 6&6 &7 \end{bmatrix}$

Possible Answers:

$\left \langle 2,1,8\right \rangle$

$\left \langle 1,-8,6\right \rangle$

$\left \langle 2,-8,9\right \rangle$

$\left \langle 1,8,6\right \rangle$

Correct answer:

$\left \langle 1,-8,6\right \rangle$

Explanation:

To find the determinant of the 3x3 matrix $\begin{bmatrix} i& j& k\\ a&b &c \\ x&y &z \end{bmatrix}$ , we use the formula:

determinant=i(bz-yc)-j(az-xc)+k(ay-xb) . Using the matrix from the problem statement, we get:

$i(7-6)-j(14-6)+k(12-6)=\left \langle 1,-8,6\right \rangle$

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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 #721 : Vectors And Vector Operations

Example Question #722 : Vectors And Vector Operations

Example Question #723 : Vectors And Vector Operations

Example Question #724 : Vectors And Vector Operations

Example Question #725 : Vectors And Vector Operations

Example Question #726 : Vectors And Vector Operations

Example Question #727 : Vectors And Vector Operations

Example Question #726 : Vectors And Vector Operations

Example Question #729 : Vectors And Vector Operations

Example Question #730 : Vectors And Vector Operations

All Calculus 3 Resources

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