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Calculus 2 : Parametric, Polar, and Vector

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9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #272 : Algebra

Given points $\displaystyle (1,1,-1)$ and $\displaystyle (0,1,-1)$, what is the vector form of the distance between the points?

Possible Answers:

$\left \langle -1,0,0\right \rangle$ $\displaystyle \left \langle -1,0,0\right \rangle$

$\left \langle 0,0,-1\right \rangle$ $\displaystyle \left \langle 0,0,-1\right \rangle$

$\left \langle 0,-1,0\right \rangle$ $\displaystyle \left \langle 0,-1,0\right \rangle$

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

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

Correct answer:

$\left \langle -1,0,0\right \rangle$ $\displaystyle \left \langle -1,0,0\right \rangle$

Explanation:

In order to derive the vector form of the distance between two points, we must find the difference between the , , and elements of the points. That is, for any point $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , the distance is the vector $v=\left \langle b_1-a_1,b_2-a_2,b_3-a_3\right \rangle$ .

Subbing in our original points $\displaystyle (1,1,-1)$ and $\displaystyle (0,1,-1)$, we get:

$v=\left \langle 0-1,1-1,-1-(-1)\right \rangle$ $\displaystyle v=\left \langle 0-1,1-1,-1-(-1)\right \rangle$

$v=\left \langle -1,0,0\right \rangle$ $\displaystyle v=\left \langle -1,0,0\right \rangle$

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Example Question #273 : Algebra

What is the vector form of 7i+6j-k $\displaystyle 7i+6j-k$?

Possible Answers:

$\left \langle 7,-6,-1\right \rangle$ $\displaystyle \left \langle 7,-6,-1\right \rangle$

$\left \langle 7,6,-1\right \rangle$ $\displaystyle \left \langle 7,6,-1\right \rangle$

$\left \langle 7,6,1\right \rangle$ $\displaystyle \left \langle 7,6,1\right \rangle$

$\left \langle 7,-6,1\right \rangle$ $\displaystyle \left \langle 7,-6,1\right \rangle$

$\left \langle -7,-6,-1\right \rangle$ $\displaystyle \left \langle -7,-6,-1\right \rangle$

Correct answer:

$\left \langle 7,6,-1\right \rangle$ $\displaystyle \left \langle 7,6,-1\right \rangle$

Explanation:

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients. That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ . So for 7i+6j-k $\displaystyle 7i+6j-k$, we can derive the vector form $\left \langle 7,6,-1\right \rangle$ $\displaystyle \left \langle 7,6,-1\right \rangle$.

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Example Question #274 : Algebra

What is the vector form of -i+j+k $\displaystyle -i+j+k$?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients. That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ . So for -i+j+k $\displaystyle -i+j+k$, we can derive the vector form $\left \langle -1,1,1\right \rangle$ $\displaystyle \left \langle -1,1,1\right \rangle$.

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Example Question #101 : Linear Algebra

Given points $\displaystyle (10,8,6)$ and $\displaystyle (7,9,11)$, what is the vector form of the distance between the points?

Possible Answers:

$\left \langle -3,-1,-5\right \rangle$ $\displaystyle \left \langle -3,-1,-5\right \rangle$

$\left \langle 3,1,-5\right \rangle$ $\displaystyle \left \langle 3,1,-5\right \rangle$

$\left \langle 3,-1,5\right \rangle$ $\displaystyle \left \langle 3,-1,5\right \rangle$

$\left \langle -3,1,5\right \rangle$ $\displaystyle \left \langle -3,1,5\right \rangle$

$\left \langle 3,-1,-5\right \rangle$ $\displaystyle \left \langle 3,-1,-5\right \rangle$

Correct answer:

$\left \langle -3,1,5\right \rangle$ $\displaystyle \left \langle -3,1,5\right \rangle$

Explanation:

Subbing in our original points $\displaystyle (10,8,6)$ and $\displaystyle (7,9,11)$, we get:

$v=\left \langle 7-10,9-8,11-6\right \rangle$ $\displaystyle v=\left \langle 7-10,9-8,11-6\right \rangle$

$v=\left \langle -3,1,5\right \rangle$ $\displaystyle v=\left \langle -3,1,5\right \rangle$

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Example Question #102 : Linear Algebra

Given points $\displaystyle (1,9,-1)$ and (2,2,7) $\displaystyle (2,2,7)$, what is the vector form of the distance between the points?

Possible Answers:

$\left \langle 1,7,-8\right \rangle$ $\displaystyle \left \langle 1,7,-8\right \rangle$

$\left \langle 1,-7,8\right \rangle$ $\displaystyle \left \langle 1,-7,8\right \rangle$

$\left \langle -1,-7,8\right \rangle$ $\displaystyle \left \langle -1,-7,8\right \rangle$

$\left \langle -1,7,8\right \rangle$ $\displaystyle \left \langle -1,7,8\right \rangle$

$\left \langle 1,7,8\right \rangle$ $\displaystyle \left \langle 1,7,8\right \rangle$

Correct answer:

$\left \langle 1,-7,8\right \rangle$ $\displaystyle \left \langle 1,-7,8\right \rangle$

Explanation:

Subbing in our original points $\displaystyle (1,9,-1)$ and (2,2,7) $\displaystyle (2,2,7)$ we get:

$v=\left \langle 2-1,2-9,7-(-1)\right \rangle$ $\displaystyle v=\left \langle 2-1,2-9,7-(-1)\right \rangle$

$v=\left \langle 1,-7,8\right \rangle$ $\displaystyle v=\left \langle 1,-7,8\right \rangle$

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Example Question #103 : Linear Algebra

What is the vector form of $\displaystyle 2i-18j+3k$?

Possible Answers:

$\left \langle 2,18,3\right \rangle$ $\displaystyle \left \langle 2,18,3\right \rangle$

$\left \langle 2,-18,3\right \rangle$ $\displaystyle \left \langle 2,-18,3\right \rangle$

$\left \langle -2,-18,-3\right \rangle$ $\displaystyle \left \langle -2,-18,-3\right \rangle$

$\left \langle 2,18,-3\right \rangle$ $\displaystyle \left \langle 2,18,-3\right \rangle$

$\left \langle -2,18,3\right \rangle$ $\displaystyle \left \langle -2,18,3\right \rangle$

Correct answer:

$\left \langle 2,-18,3\right \rangle$ $\displaystyle \left \langle 2,-18,3\right \rangle$

Explanation:

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients.

That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ .

So for $\displaystyle 2i-18j+3k$ , we can derive the vector form $\left \langle 2,-18,3\right \rangle$ $\displaystyle \left \langle 2,-18,3\right \rangle$.

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Example Question #104 : Linear Algebra

What is the vector form of $\displaystyle -k$?

Possible Answers:

$\left \langle 0,-1,0\right \rangle$ $\displaystyle \left \langle 0,-1,0\right \rangle$

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

$\left \langle -1,0,0\right \rangle$ $\displaystyle \left \langle -1,0,0\right \rangle$

$\left \langle 0,0,-1\right \rangle$ $\displaystyle \left \langle 0,0,-1\right \rangle$

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

Correct answer:

$\left \langle 0,0,-1\right \rangle$ $\displaystyle \left \langle 0,0,-1\right \rangle$

Explanation:

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients.

That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ .

So for $\displaystyle -k$ , we can derive the vector form $\left \langle 0,0,-1\right \rangle$ $\displaystyle \left \langle 0,0,-1\right \rangle$.

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Example Question #105 : Linear Algebra

Given points $\displaystyle (2,0,-10)$ and $\displaystyle (-5,1,6)$, what is the vector form of the distance between the points?

Possible Answers:

$\left \langle 7,1,16\right \rangle$ $\displaystyle \left \langle 7,1,16\right \rangle$

$\left \langle -7,1,16\right \rangle$ $\displaystyle \left \langle -7,1,16\right \rangle$

$\left \langle 7,1,-16\right \rangle$ $\displaystyle \left \langle 7,1,-16\right \rangle$

$\left \langle -7,-1,16\right \rangle$ $\displaystyle \left \langle -7,-1,16\right \rangle$

$\left \langle 7,-1,16\right \rangle$ $\displaystyle \left \langle 7,-1,16\right \rangle$

Correct answer:

$\left \langle -7,1,16\right \rangle$ $\displaystyle \left \langle -7,1,16\right \rangle$

Explanation:

Subbing in our original points $\displaystyle (2,0,-10)$ and $\displaystyle (-5,1,6)$, we get:

$v=\left \langle -5-2,1-0,6-(-10)\right \rangle$ $\displaystyle v=\left \langle -5-2,1-0,6-(-10)\right \rangle$

$v=\left \langle -7,1,16\right \rangle$ $\displaystyle v=\left \langle -7,1,16\right \rangle$

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Example Question #106 : Linear Algebra

Given points $\displaystyle (5,-5,5)$ and $\displaystyle (-4,4,-4)$, what is the vector form of the distance between the points?

Possible Answers:

$\left \langle -9,-9,-9\right \rangle$ $\displaystyle \left \langle -9,-9,-9\right \rangle$

$\left \langle 9,9,9\right \rangle$ $\displaystyle \left \langle 9,9,9\right \rangle$

$\left \langle 9,-9,-9\right \rangle$ $\displaystyle \left \langle 9,-9,-9\right \rangle$

$\left \langle -9,-9,9\right \rangle$ $\displaystyle \left \langle -9,-9,9\right \rangle$

$\left \langle -9,9,-9\right \rangle$ $\displaystyle \left \langle -9,9,-9\right \rangle$

Correct answer:

$\left \langle -9,9,-9\right \rangle$ $\displaystyle \left \langle -9,9,-9\right \rangle$

Explanation:

Subbing in our original points $\displaystyle (5,-5,5)$ and $\displaystyle (-4,4,-4)$, we get:

$v=\left \langle -4-5,4-(-5),-4-5\right \rangle$ $\displaystyle v=\left \langle -4-5,4-(-5),-4-5\right \rangle$

$v=\left \langle -9,9,-9\right \rangle$ $\displaystyle v=\left \langle -9,9,-9\right \rangle$

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Example Question #107 : Linear Algebra

What is the vector form of $\displaystyle 8i-9j+10k$?

Possible Answers:

$\left \langle 8,9,10\right \rangle$ $\displaystyle \left \langle 8,9,10\right \rangle$

$\left \langle -8,-9,10\right \rangle$ $\displaystyle \left \langle -8,-9,10\right \rangle$

$\left \langle -8,9,10\right \rangle$ $\displaystyle \left \langle -8,9,10\right \rangle$

$\left \langle 8,9,-10\right \rangle$ $\displaystyle \left \langle 8,9,-10\right \rangle$

$\left \langle 8,-9,10\right \rangle$ $\displaystyle \left \langle 8,-9,10\right \rangle$

Correct answer:

$\left \langle 8,-9,10\right \rangle$ $\displaystyle \left \langle 8,-9,10\right \rangle$

Explanation:

In order to derive the vector form, we must map the , , -coordinates to their corresponding , , and coefficients. That is, given $a_{1}i+a_{2}j+a_{3}k$ , the vector form is $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ . So for $\displaystyle 8i-9j+10k$, we can derive the vector form $\left \langle 8,-9,10\right \rangle$ $\displaystyle \left \langle 8,-9,10\right \rangle$.

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Calculus 2 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #272 : Algebra

Example Question #273 : Algebra

Example Question #274 : Algebra

Example Question #101 : Linear Algebra

Example Question #102 : Linear Algebra

Example Question #103 : Linear Algebra

Example Question #104 : Linear Algebra

Example Question #105 : Linear Algebra

Example Question #106 : Linear Algebra

Example Question #107 : Linear Algebra

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