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Calculus 3 : Calculus 3

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 #11 : Distance Between Vectors

Find the distance between the vectors $\left \langle 2,2,4\right \rangle$ and $\left \langle 3,0,9\right \rangle$ .

Possible Answers:

$\sqrt{30}$

Correct answer:

$\sqrt{30}$

Explanation:

To find the distance between 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 $d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$ . Using the vectors given, we get $\sqrt{(2-3)^2+(2-0)^2+(4-9)^2}=\sqrt{1+4+25}=\sqrt{30}$

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Example Question #11 : Distance Between Vectors

Find the distance between the vectors $\left \langle 5,3,8\right \rangle$ and $\left \langle 3,0,9\right \rangle$

Possible Answers:

$\sqrt{11}$

$\sqrt{13}$

$\sqrt{14}$

Correct answer:

$\sqrt{14}$

Explanation:

The formula for the distance between vectors $a=\left \langle a,b,c\right \rangle$ and $b=\left \langle x,y,z\right \rangle$ is $d=\sqrt{(a-x)^2+(b-y)^2+(c-z)^2}$ . Using the vectors from the problem statement, we get $\sqrt{(5-3)^2+(3-0)^2+(8-9)^2}=\sqrt{14}$

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Example Question #12 : Distance Between Vectors

Find the distance between the vectors $\left \langle 4,11,10\right \rangle$ and $\left \langle 4,1,0\right \rangle$

Possible Answers:

$\sqrt{200}$

Correct answer:

$\sqrt{200}$

Explanation:

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Example Question #14 : Distance Between Vectors

Find the distance between the vectors $\left \langle 2,2,4\right \rangle$ and $\left \langle 1,-4,1\right \rangle$

Possible Answers:

$\sqrt{10}$

$\sqrt{46}$

Correct answer:

$\sqrt{46}$

Explanation:

To find the distance between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula $d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$ . Plugging in the vectors from the problem statement, we get $d=\sqrt{(2-1)^2+(2+4)^2+(4-1)^2}=\sqrt{1+36+9}=\sqrt{46}$

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Example Question #15 : Distance Between Vectors

Find the distance between the vectors $\left \langle -3,1,-4\right \rangle$ and $\left \langle 2,-5,0\right \rangle$

Possible Answers:

$\sqrt{66}$

$\sqrt{77}$

Correct answer:

$\sqrt{77}$

Explanation:

o find the distance between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula $d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$ . Plugging in the vectors from the problem statement, we get $d=\sqrt{(-3-2)^2+(1+5)^2+(-4-0)^2}=\sqrt{25+36+16}=\sqrt{77}$

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Example Question #16 : Distance Between Vectors

Find the distance between the vectors $\left \langle 13,7,12\right \rangle$ and $\left \langle 8,7,10\right \rangle$

Possible Answers:

$\sqrt{20}$

$\sqrt{29}$

Correct answer:

$\sqrt{29}$

Explanation:

To find the distance between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula

$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$

Using the vectors from the problem statement, we get

$d=\sqrt{(13-8)^2+(7-7)^2+(12-10)^2}=\sqrt{29}$

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Example Question #17 : Distance Between Vectors

Find the distance between the vectors $\left \langle 10,5,6\right \rangle$ and $\left \langle 9,8,7\right \rangle$

Possible Answers:

$\sqrt{11}$

$\sqrt{12}$

Correct answer:

$\sqrt{11}$

Explanation:

To find the distance between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula

$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$

Using the vectors from the problem statement, we get

$d=\sqrt{(10-9)^2+(5-8)^2+(6-7)^2}=\sqrt{11}$

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Example Question #13 : Distance Between Vectors

Find the distance between the vectors $\left \langle 3,4,5\right \rangle$ and $\left \langle 1,5,2\right \rangle$

Possible Answers:

$\sqrt{5}$

$\sqrt{15}$

$\sqrt{14}$

Correct answer:

$\sqrt{14}$

Explanation:

To find the distance between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula

$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$

Using the vectors from the problem statement, we get

$d=\sqrt{(3-1)^2+(4-5)^2+(5-2)^2}=\sqrt{14}$

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Example Question #14 : Distance Between Vectors

Find the distance between the vectors $\left \langle 0,7,11\right \rangle$ and $\left \langle -3,-2,9\right \rangle$

Possible Answers:

$\sqrt{66}$

$\sqrt{82}$

$\sqrt{94}$

Correct answer:

$\sqrt{94}$

Explanation:

To find the distance between two vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula

$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$

Using the vectors from the problem statement, we get

$d=\sqrt{(0+3)^2+(7+2)^2+(11-9)^2}=\sqrt{94}$

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Example Question #20 : Distance Between Vectors

Find the distance between the two vectors u and v:

$\mathbf{u}=13\mathbf{i}+8\mathbf{j}+2\sqrt{2}\mathbf{k}$

$\mathbf{v}=9\mathbf{i}+3\mathbf{j}$

Possible Answers:

Correct answer:

Explanation:

The Euclidean distance between two vectors:

$\mathbf{u}=u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k}$ and $\mathbf{v}=v_x\mathbf{i}+v_y\mathbf{j}+v_z\mathbf{k}$

is given by:

$d=\sqrt{(u_x-v_x)^2+(u_y-v_y)^2+(u_z-v_z)^2}$

For the given vectors, this results in the distance:

$d=\sqrt{(13-9)^2+(8-3)^2+(2\sqrt{2}-0)^2}$

$d=\sqrt{4^2+5^2+(2\sqrt{2})^2}=\sqrt{16+25+8}=\sqrt{49}=7$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #11 : Distance Between Vectors

Example Question #11 : Distance Between Vectors

Example Question #12 : Distance Between Vectors

Example Question #14 : Distance Between Vectors

Example Question #15 : Distance Between Vectors

Example Question #16 : Distance Between Vectors

Example Question #17 : Distance Between Vectors

Example Question #13 : Distance Between Vectors

Example Question #14 : Distance Between Vectors

Example Question #20 : Distance Between Vectors

All Calculus 3 Resources

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