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Calculus 3 : Vectors and Vector Operations

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

Example Questions

Example Question #81 : Normal Vectors

Find the normal vector to the following vectors:

$\left \langle 8, 1, -4\right \rangle, \left \langle -1, 3, 4\right \rangle$

Possible Answers:

$\left \langle 16, 28, 23\right \rangle$

$\left \langle 16, -28, 23\right \rangle$

$\left \langle -8, 36, 25\right \rangle$

Correct answer:

$\left \langle 16, -28, 23\right \rangle$

Explanation:

To find the normal vector, we must take the cross product of the two vectors.

Now, we can write the determinant in order to take the cross product of the two vectors:
$\begin{vmatrix} \hat{i}& \hat{j}&\hat{k} \\ 8&1 &-4 \\ -1&3 &4 \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:

$4\hat{i}+24\hat{k}+4\hat{j}-(\hat{k}-12\hat{i}+32\hat{j})=16\hat{i}-28\hat{j}+23\hat{k}=\left \langle 16, -28, 23\right \rangle$

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Example Question #88 : Normal Vectors

Determine the vector normal to the plane created by the following two vectors:

$\left \langle 10, 2, 4\right \rangle, \left \langle 11, 2, 6\right \rangle$

Possible Answers:

$\left \langle 20, 104, 42\right \rangle$

$\left \langle -16, 4, -2\right \rangle$

$\left \langle 4, 16, 2\right \rangle$

$\left \langle 4, -16, -2\right \rangle$

Correct answer:

$\left \langle 4, -16, -2\right \rangle$

Explanation:

The normal vector to a plane is given by the cross product of two vectors in that plane.

So, we write the determinant in order to take the cross product of the two vectors:

$\begin{vmatrix} \hat{i}& \hat{j}&\hat{k} \\ 10&2 &4 \\ 11&2 &6 \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

$12\hat{i}+20\hat{k}+44\hat{j}-(22\hat{k}+8\hat{i}+60\hat{j})=4\hat{i}-16\hat{j}-2\hat{k}=\left \langle 4, -16, -2\right \rangle$

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Example Question #89 : Normal Vectors

Find the normal vector to the plane containing the vectors $\left \langle3,4,-6 \right \rangle$ and $\left \langle 0,5,3\right \rangle$

Possible Answers:

$\left \langle 20,-9,15\right \rangle$

$\left \langle 42,-9,15\right \rangle$

$\left \langle 23,9,15\right \rangle$

$\left \langle 23,-9,11\right \rangle$

Correct answer:

$\left \langle 42,-9,15\right \rangle$

Explanation:

To find the normal vector to a plane containing vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you take the cross product of the two vectors.

To find the cross product between the two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you take the determinant of the 3x3 matrix

$\begin{bmatrix} \hat{i}&\hat{j} &\hat{k} \\ a_1& a_2&a_3 \\ b_1&b_2 &b_3 \end{bmatrix}$ . $determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using the vectors from the problem statement, we get

$determinant=\hat{i}(12+30)-\hat{j}(9-0)+\hat{k}(15-0)=\left \langle 42,-9,15\right \rangle$

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Example Question #90 : Normal Vectors

Find the normal vector to the plane containing the vectors $\left \langle0,7,5 \right \rangle$ and $\left \langle -9,-1,-4\right \rangle$

Possible Answers:

$\left \langle -20,-45,61\right \rangle$

$\left \langle -23,-45,63\right \rangle$

$\left \langle -23,45,63\right \rangle$

$\left \langle -23,-44,63\right \rangle$

Correct answer:

$\left \langle -23,-45,63\right \rangle$

Explanation:

To find the normal vector to a plane containing vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you take the cross product of the two vectors.

To find the cross product between the two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you take the determinant of the 3x3 matrix

$\begin{bmatrix} \hat{i}&\hat{j} &\hat{k} \\ a_1& a_2&a_3 \\ b_1&b_2 &b_3 \end{bmatrix}$ . $determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using the vectors from the problem statement, we get

$determinant=\hat{i}(-28+5)-\hat{j}(0+45)+\hat{k}(0+63)=\left \langle -23,-45,63\right \rangle$

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Example Question #91 : Normal Vectors

Find the normal vector to the plane containing the vectors $\left \langle0,-5,-4 \right \rangle$ and $\left \langle2,-4,7\right \rangle$

Possible Answers:

$\left \langle -50,-8,9\right \rangle$

$\left \langle -51,-5,10\right \rangle$

$\left \langle -41,-8,10\right \rangle$

$\left \langle -51,-8,10\right \rangle$

Correct answer:

$\left \langle -51,-8,10\right \rangle$

Explanation:

To find the normal vector to a plane containing vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you take the cross product of the two vectors.

To find the cross product between the two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you take the determinant of the 3x3 matrix

$\begin{bmatrix} \hat{i}&\hat{j} &\hat{k} \\ a_1& a_2&a_3 \\ b_1&b_2 &b_3 \end{bmatrix}$ . $determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using the vectors from the problem statement, we get

$determinant=\hat{i}(-35-16)-\hat{j}(0+8)+\hat{k}(0+10)=\left \langle -51,-8,10\right \rangle$

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Example Question #92 : Normal Vectors

Find the normal vector to the plane containing the vectors $\left \langle -2,0,6\right \rangle$ and $\left \langle 2,-3,-1\right \rangle$

Possible Answers:

$\left \langle 18,11,5\right \rangle$

$\left \langle 10,13,6\right \rangle$

$\left \langle 18,14,6\right \rangle$

$\left \langle 18,-14,6\right \rangle$

Correct answer:

$\left \langle 18,14,6\right \rangle$

Explanation:

To obtain the normal vector to a plane containing two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you compute the determinant of the 3x3 matrix $\begin{bmatrix} \hat{i}&\hat{j} &\hat{k} \\ a_1&a_2 &a_3 \\ b_1&b_2 & b_3 \end{bmatrix}$

$determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using this formula, we evaluate using the vectors from the problem statement:

$\hat{i}(0+18)-\hat{j}(-2-12)+\hat{k}(6-0)=\left \langle 18,14,6\right \rangle$

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Example Question #93 : Normal Vectors

Find the normal vector to the plane that contains the vectors $\left \langle 0,4,9\right \rangle$ and $\left \langle 0,7,5\right \rangle$

Possible Answers:

$\left \langle -25,0,0\right \rangle$

$\left \langle -27,0,0\right \rangle$

$\left \langle -27,3,6\right \rangle$

$\left \langle -27,1,1\right \rangle$

Correct answer:

$\left \langle -27,0,0\right \rangle$

Explanation:

$determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using this formula, we evaluate using the vectors from the problem statement:

$\hat{i}(36-63)-\hat{j}(0-0)+\hat{k}(0-0)=\left \langle -27,0,0\right \rangle$

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Example Question #94 : Normal Vectors

Find the normal vector to the plane that contains the vectors $\left \langle 5,0,-6\right \rangle$ and $\left \langle 7,0,2\right \rangle$

Possible Answers:

$\left \langle -3,16,3\right \rangle$

$\left \langle 0,51,0\right \rangle$

$\left \langle -3,12,0\right \rangle$

$\left \langle 0,52,0\right \rangle$

Correct answer:

$\left \langle 0,52,0\right \rangle$

Explanation:

$determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using this formula, we evaluate using the vectors from the problem statement:

$\hat{i}(0-0)-\hat{j}(10+42)+\hat{k}(0-0)=\left \langle 0,52,0\right \rangle$

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Example Question #95 : Normal Vectors

Find the normal vector to the plane that contains the vectors $\left \langle 2,0,9\right \rangle$ and $\left \langle -3,0,12\right \rangle$

Possible Answers:

$\left \langle 27,33,3\right \rangle$

$\left \langle 0,51,0\right \rangle$

$\left \langle -2,48,0\right \rangle$

$\left \langle 3,5,18\right \rangle$

Correct answer:

$\left \langle 0,51,0\right \rangle$

Explanation:

$determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using this formula, we evaluate using the vectors from the problem statement:

$\hat{i}(0-0)-\hat{j}(24+27)+\hat{k}(0-0)=\left \langle 0,51,0\right \rangle$

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Example Question #95 : Normal Vectors

Find the Unit Normal Vector to the given plane.

3x+3y+4z=10 .

Possible Answers:

$V_n=(\frac{1}{\sqrt{34}}, \frac{1}{\sqrt{34}}, \frac{1}{\sqrt{34}} )$

$V_n=(\frac{1}{{34}}, \frac{1}{{34}}, \frac{1}{{34}} )$

$V_n=(\frac{3}{\sqrt{34}}, \frac{3}{\sqrt{34}}, \frac{4}{\sqrt{34}} )$

V_n=(3, 3, 4 )

$V_n=(\frac{3}{{34}}, \frac{3}{{34}}, \frac{4}{{34}} )$

Correct answer:

$V_n=(\frac{3}{\sqrt{34}}, \frac{3}{\sqrt{34}}, \frac{4}{\sqrt{34}} )$

Explanation:

Recall the definition of the Unit Normal Vector.

$V_n=\frac{V}{||V||}$

Let V=(3,3,4)

$||V||=\sqrt{3^2+3^2+4^2}=\sqrt{34}$

$V_n=(\frac{3}{\sqrt{34}}, \frac{3}{\sqrt{34}}, \frac{4}{\sqrt{34}} )$

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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 #81 : Normal Vectors

Example Question #88 : Normal Vectors

Example Question #89 : Normal Vectors

Example Question #90 : Normal Vectors

Example Question #91 : Normal Vectors

Example Question #92 : Normal Vectors

Example Question #93 : Normal Vectors

Example Question #94 : Normal Vectors

Example Question #95 : Normal Vectors

Example Question #95 : Normal Vectors

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