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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

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

Example Questions

Example Question #53 : Vectors & Spaces

What is the vector form of 4i+k ?

Possible Answers:

$\left \langle 0,4,1\right \rangle$

$\left \langle 4,0,1\right \rangle$

$\left \langle -4,0,1\right \rangle$

$\left \langle 4,0,-1\right \rangle$

$\left \langle -4,0,-1\right \rangle$

Correct answer:

$\left \langle 4,0,1\right \rangle$

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 4i+k , we can derive the vector form $\left \langle 4,0,1\right \rangle$ .

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Example Question #73 : Linear Algebra

What is the vector form of 2i-3j ?

Possible Answers:

$\left \langle 0,2,-3\right \rangle$

$\left \langle 2,3,0\right \rangle$

$\left \langle- 2,3,0\right \rangle$

$\left \langle- 2,-3,0\right \rangle$

$\left \langle 2,-3,0\right \rangle$

Correct answer:

$\left \langle 2,-3,0\right \rangle$

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 2i-3j , we can derive the vector form $\left \langle 2,-3,0\right \rangle$ .

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Example Question #51 : Vectors

Find the vector in standard form if the initial point is located at (3,7) and the terminal point is located at (7,10) .

Possible Answers:

-4i-3j

4i+3j

7i+3j

i+j

Correct answer:

4i+3j

Explanation:

We must first find the vector in component form.

If the initial point is (p_1,p_2) and the terminal point is (q_1,q_2) then the component form of the vector is $\left<q_1-p_1,q_2-p_2\right>$ .

As such, the component form of the vector in the problem is $\left<7-3,10-7\right>$ $=\left<4,3\right>$

Next, any vector with component form $\left<a,b\right>$ can be written in standard form as ai+bj .

Hence, the vector in standard form is

4i+3j

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Example Question #52 : Vector Form

What is the vector form of 3i-5j+7k ?

Possible Answers:

$\left \langle 3,-5,7\right \rangle$

$\left \langle -3,-5,-7\right \rangle$

$\left \langle -3,-5,7\right \rangle$

$\left \langle 3,5,-7\right \rangle$

$\left \langle -3,5,7\right \rangle$

Correct answer:

$\left \langle 3,-5,7\right \rangle$

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 3i-5j+7k , we can derive the vector form $\left \langle 3,-5,7\right \rangle$ .

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Example Question #432 : Gre Subject Test: Math

What is the vector form of ?

Possible Answers:

$\left \langle 0,0,7\right \rangle$

$\left \langle -7,0,0\right \rangle$

$\left \langle 0,0,-7\right \rangle$

$\left \langle 7,0,0\right \rangle$

$\left \langle 0,7,0,\right \rangle$

Correct answer:

$\left \langle 7,0,0\right \rangle$

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 , we can derive the vector form $\left \langle 7,0,0\right \rangle$ .

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Example Question #51 : Vectors

Given points (9,4,-1) and (5,2,7) , what is the vector form of the distance between the points?

Possible Answers:

$\left \langle 4,-2,8\right \rangle$

$\left \langle -4,-2,8\right \rangle$

$\left \langle -4,-2,-8\right \rangle$

$\left \langle 4,2,8\right \rangle$

$\left \langle -4,2,8\right \rangle$

Correct answer:

$\left \langle -4,-2,8\right \rangle$

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 (9,4,-1) and (5,2,7) , we get:

$v=\left \langle 5-9,2-4,7-(-1)\right \rangle$

$v=\left \langle -4,-2,8\right \rangle$

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Example Question #61 : Vectors

What is the vector form of -6j ?

Possible Answers:

$\left \langle 0,-6,0 \right \rangle$

$\left \langle 0,0,-6 \right \rangle$

$\left \langle 6, 0,0 \right \rangle$

$\left \langle 0,6,0 \right \rangle$

$\left \langle -6, 0,0 \right \rangle$

Correct answer:

$\left \langle 0,-6,0 \right \rangle$

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 -6j , we can derive the vector form $\left \langle 0,-6,0 \right \rangle$ .

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Example Question #82 : Linear Algebra

Given points (0,1,3) and (5,7,4) , what is the vector form of the distance between the points?

Possible Answers:

$\left \langle 5,-6,1\right \rangle$

$\left \langle 5,6,1\right \rangle$

$\left \langle -5,-6,-1\right \rangle$

$\left \langle -5,6,1\right \rangle$

$\left \langle -5,-6,1\right \rangle$

Correct answer:

$\left \langle 5,6,1\right \rangle$

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 (0,1,3) and (5,7,4) , we get:

$v=\left \langle 5-0,7-1,4-3\right \rangle$

$v=\left \langle 5,6,1\right \rangle$

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Example Question #63 : Vectors & Spaces

Given points (2,4,5) and (9,1,0) , what is the vector form of the distance between the points?

Possible Answers:

$\left \langle 7,3,5\right \rangle$

$\left \langle -7,-3,-5\right \rangle$

$\left \langle 7,-3,-5\right \rangle$

$\left \langle -7,3,-5\right \rangle$

$\left \langle 7,3,-5\right \rangle$

Correct answer:

$\left \langle 7,-3,-5\right \rangle$

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 (2,4,5) and (9,1,0) , we get:

$v=\left \langle 9-2,1-4,0-5\right \rangle$

$v=\left \langle 7,-3,-5\right \rangle$

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Example Question #61 : Vector Form

Given points (0,0,-4) and (4,9,-1) , what is the vector form of the distance between the points?

Possible Answers:

$\left \langle -4,9,-3\right \rangle$

$\left \langle 4,9,-3\right \rangle$

$\left \langle 4,9,3\right \rangle$

$\left \langle 4,-9,3\right \rangle$

$\left \langle -4,9,3\right \rangle$

Correct answer:

$\left \langle 4,9,3\right \rangle$

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 (0,0,-4) and (4,9,-1) , we get:

$v=\left \langle 4-0,9-0,-1+4\right \rangle$

$v=\left \langle 4,9,3\right \rangle$

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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #53 : Vectors & Spaces

Example Question #73 : Linear Algebra

Example Question #51 : Vectors

Example Question #52 : Vector Form

Example Question #432 : Gre Subject Test: Math

Example Question #51 : Vectors

Example Question #61 : Vectors

Example Question #82 : Linear Algebra

Example Question #63 : Vectors & Spaces

Example Question #61 : Vector Form

All Calculus 2 Resources

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