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Calculus 3 : Distance between Vectors

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 #21 : Distance Between Vectors

Find the distance between the vectors $\left \langle 3,-6,1\right \rangle$ and $\left \langle 2,5,5\right \rangle$

Possible Answers:

$\sqrt{138}$

$\sqrt{131}$

$\sqrt{115}$

$\sqrt{120}$

Correct answer:

$\sqrt{138}$

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 following formula:

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

Therefore, we get

$d=\sqrt{(3-2)^2+(-6-5)^2+(1-5)^2}=\sqrt{138}$

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Example Question #22 : Distance Between Vectors

Find the distance between the vectors $\left \langle 3,0,6\right \rangle$ and $\left \langle -4,1,3\right \rangle$

Possible Answers:

$\sqrt{58}$

$\sqrt{60}$

$\sqrt{57}$

$\sqrt{59}$

Correct answer:

$\sqrt{59}$

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 following formula:

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

Therefore, we get

$d=\sqrt{(3+4)^2+(0-1)^2+(6-3)^2}=\sqrt{59}$

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Example Question #23 : Distance Between Vectors

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

Possible Answers:

$\sqrt{13}$

$\sqrt{10}$

$\sqrt{11}$

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 apply the following formula:

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

Using the vectors form the problem statement and applying, we get

$d=\sqrt{(3-2)^2+(0-1)^2+(4-1)^2}=\sqrt{11}$

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Example Question #24 : Distance Between Vectors

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

Possible Answers:

$\sqrt{19}$

$\sqrt{18}$

$\sqrt{30}$

$\sqrt{12}$

Correct answer:

$\sqrt{30}$

Explanation:

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

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

Using the vectors form the problem statement and applying, we get

$d=\sqrt{(5-3)^2+(5-0)^2+(2-1)^2}=\sqrt{30}$

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Example Question #25 : Distance Between Vectors

Find the distance between the vectors $\left \langle 2,1,1\right \rangle$ and $\left \langle 3,6,5\right \rangle$

Possible Answers:

$\sqrt{42}$

$\sqrt{41}$

$\sqrt{45}$

$\sqrt{37}$

Correct answer:

$\sqrt{42}$

Explanation:

To find the distance between two vectors $\left \langle x_1,y_1,z_1\right \rangle$ and $\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula $d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$

Applying to the vectors from the problem statement, we get

$d=\sqrt{(2-3)^2+(1-6)^2+(1-5)^2}=\sqrt{42}$

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Example Question #26 : Distance Between Vectors

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

Possible Answers:

Correct answer:

Explanation:

To find the distance between two vectors $\left \langle x_1,y_1,z_1\right \rangle$ and $\left \langle x_2,y_2,z_2\right \rangle$ , we use the formula $d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$

Applying to the vectors from the problem statement, we get

$d=\sqrt{(0-0)^2+(3-3)^2+(1-5)^2}=\sqrt{16}=4$

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Example Question #21 : Distance Between Vectors

Find the distance between the vectors $\left \langle 9,3,8\right \rangle$ and $\left \langle 2,-6,1\right \rangle$

Possible Answers:

$\sqrt{107}$

$\sqrt{105}$

$\sqrt{117}$

$\sqrt{111}$

Correct answer:

$\sqrt{107}$

Explanation:

To find the distance between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we use the formula

$d=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2}$

Using the vectors from the problem statement, we get

$d=\sqrt{(9-2)^2+(3-6)^2+(8-1)^2}=\sqrt{107}$

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Example Question #22 : Distance Between Vectors

Find the distance between the vectors $\left \langle 4,5,-9\right \rangle$ and $\left \langle 5,2,-5\right \rangle$

Possible Answers:

$\sqrt{26}$

$\sqrt{31}$

$\sqrt{30}$

Correct answer:

$\sqrt{26}$

Explanation:

To find the distance between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we use the formula

$d=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2}$

Using the vectors from the problem statement, we get

$d=\sqrt{(4-5)^2+(5-2)^2+(-9+5)^2}=\sqrt{26}$

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Example Question #21 : Distance Between Vectors

Find the distance between the vectors $\left \langle 7,0,7\right \rangle$ and $\left \langle 5,7,-5\right \rangle$

Possible Answers:

$\sqrt{155}$

$\sqrt{197}$

$\sqrt{191}$

$\sqrt{173}$

Correct answer:

$\sqrt{197}$

Explanation:

To find the distance between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we use the formula

$d=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2}$

Using the vectors from the problem statement, we get

$d=\sqrt{(7-5)^2+(0-7)^2+(7+5)^2}=\sqrt{197}$

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Example Question #30 : Distance Between Vectors

Find the distance between the vectors $\left \langle 2,-5,4\right \rangle$ and $\left \langle -8,9,3\right \rangle$

Possible Answers:

$\sqrt{302}$

$\sqrt{299}$

$\sqrt{297}$

$\sqrt{155}$

Correct answer:

$\sqrt{297}$

Explanation:

To find the distance between 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 use the formula

$d=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2}$

Using the vectors from the problem statement, we get

$d=\sqrt{(2+8)^2+(-5-9)^2+(4-3)^2}=\sqrt{297}$

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Calculus 3 : Distance between Vectors

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #21 : Distance Between Vectors

Example Question #22 : Distance Between Vectors

Example Question #23 : Distance Between Vectors

Example Question #24 : Distance Between Vectors

Example Question #25 : Distance Between Vectors

Example Question #26 : Distance Between Vectors

Example Question #21 : Distance Between Vectors

Example Question #22 : Distance Between Vectors

Example Question #21 : Distance Between Vectors

Example Question #30 : Distance Between Vectors

All Calculus 3 Resources

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