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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

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All Calculus 2 Resources

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

Example Questions

Example Question #1 : Vector Form

Express -10j in vector form.

Possible Answers:

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

$\left \langle -10,0,0\right \rangle$

$\left \langle 1,-10,1\right \rangle$

$\left \langle 0,-10,0\right \rangle$

$\left \langle 0,0,-10\right \rangle$

Correct answer:

$\left \langle 0,-10,0\right \rangle$

Explanation:

The x,y, and z of a vector is represented in the order of i, j, and k, respectively. Use the coefficients of i,j, and k to write the vector form.

$-10j=0i-10j+0k = \left \langle 0,-10,0\right \rangle$

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Example Question #2 : Vector Form

Find the vector form of (6,3,1) to (0,1,3) .

Possible Answers:

$\overrightarrow{v}=[-6,-2,-2]$

$\overrightarrow{v}=[6,-2,-2]$

$\overrightarrow{v}=[6,2,-2]$

$\overrightarrow{v}=[6,2,2]$

$\overrightarrow{v}=[6,2,-3]$

Correct answer:

$\overrightarrow{v}=[6,2,-2]$

Explanation:

When we are trying to find the vector form we need to remember the formula which states to take the difference between the ending and starting point.

Thus we would get:

Given (a,b,c) and (d,e,f)

$\overrightarrow{v}=[d-a, e-b, f-c]$

In our case we have ending point at (6,3,1) and our starting point at (0,1,3) .

Therefore we would set up the following and simplify.

$\overrightarrow{v}=[6-0,3-1,1-3]=[6,2,-2]$

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Example Question #25 : Linear Algebra

Find the dot product of the 2 vectors. $\left \langle a,2a\right \rangle \cdot \left \langle a,a\right \rangle$

Possible Answers:

$\left \langle a^2,2a^2\right \rangle$

$\left \langle 3a^2,3a^2\right \rangle$

$\left \langle a,2a\right \rangle$

3a^2

Correct answer:

3a^2

Explanation:

The dot product will give a single value answer, and not a vector as a result.

To find the dot product, use the following formula:

$\left \langle x_1,x_2\right \rangle \cdot \left \langle y_1,y_2\right \rangle=(x_1)(y_1)+(x_2)(y_2)$

$\left \langle a,2a\right \rangle \cdot \left \langle a,a\right \rangle = (a)(a)+(2a)(a)=a^2+2a^2=3a^2$

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Example Question #2 : Vectors & Spaces

Assume that Billy fired himself out of a circus cannon at a velocity of $25 \:mph$ at an elevation angle of degrees. Write this in vector component form.

Possible Answers:

$\left \langle22.658,10.565 \right \rangle$

$\left \langle 25.25,25.25\right \rangle$

$\left \langle23.097, 9.567 \right \rangle$

$\left \langle 50,50\right \rangle$

$\left \langle 24.78,-3.309\right \rangle$

Correct answer:

$\left \langle22.658,10.565 \right \rangle$

Explanation:

The firing of the cannon has both x and y components.

Write the formula that distinguishes the x and y direction and substitute.

$\left \langle x,y\right \rangle= \left \langle vcos(\theta),vsin(\theta)\right \rangle$

Ensure that the calculator is in degree mode before you solve.

$\left \langle 25cos25,25sin25\right \rangle = \left \langle22.658,10.565 \right \rangle$

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Example Question #1 : Vectors

Compute: $2\vec{a}-6\vec{b}$ given the following vectors. $\vec{a}= \left \langle 1,3\right \rangle$ and $\vec{b}= \left \langle 1\right \rangle$ .

Possible Answers:

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

${}\left< \sqrt{13} \right \rangle$

The answer does not exist.

$\left \langle 1,2\right \rangle$

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

Correct answer:

The answer does not exist.

Explanation:

The dimensions of the vectors are mismatched.

Since vector $\vec{a}$ does not have the same dimensions as $\vec{b}$ , the answer for $2\vec{a}-6\vec{b}$ cannot be solved.

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Example Question #1 : Vector Form

What is the vector form of 2i-10k ?

Possible Answers:

$\left \langle 2,0,-10\right \rangle$

$\left \langle -10,2,0\right \rangle$

$\left \langle 2,-10,0\right \rangle$

Correct answer:

$\left \langle 2,0,-10\right \rangle$

Explanation:

To find the vector form of 2i-10k , we must map the coefficients of , , and to their corresponding , , and coordinates. Thus, 2i-10k becomes $\left \langle 2,0,-10\right \rangle$ .

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Example Question #1 : Vector Form

Express -2i+7j-10k in vector form.

Possible Answers:

$\left \langle 2,-7,10\right \rangle$

$\left \langle -2,-7,-10\right \rangle$

$\left \langle 2,7,-10\right \rangle$

$\left \langle -2,7,-10\right \rangle$

$\left \langle 2,7,10\right \rangle$

Correct answer:

$\left \langle -2,7,-10\right \rangle$

Explanation:

In order to express -2i+7j-10k in vector form, we must use the coefficients of i, j, and to represent the -, -, and -coordinates of the vector.

Therefore, its vector form is

$\left \langle -2,7,-10\right \rangle$ .

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Example Question #1 : Vector Form

Express i+5k in vector form.

Possible Answers:

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

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

$\left \langle -1,0,5\right \rangle$

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

$\left \langle 1,0,5\right \rangle$

Correct answer:

$\left \langle 1,0,5\right \rangle$

Explanation:

In order to express i+5k in vector form, we must use the coefficients of i, j, and to represent the -, -, and -coordinates of the vector.

Therefore, its vector form is

$\left \langle 1,0,5\right \rangle$ .

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Example Question #1 : Vector Form

Express -j+7k in vector form.

Possible Answers:

$\left \langle -1,7,0\right \rangle$

$\left \langle -1,0,7\right \rangle$

$\left \langle 0,-1,7\right \rangle$

None of the above

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

Correct answer:

$\left \langle 0,-1,7\right \rangle$

Explanation:

In order to express -j+7k in vector form, we will need to map its , , and coefficients to its -, -, and -coordinates.

Thus, its vector form is

$\left \langle 0,-1,7\right \rangle$ .

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Example Question #11 : Vector Form

Express 5i+9j-2k in vector form.

Possible Answers:

$\left \langle 5,9,-2\right \rangle$

$\left \langle 9,5,-2\right \rangle$

$\left \langle -2,5,9\right \rangle$

$\left \langle 9,-2,5\right \rangle$

None of the above

Correct answer:

$\left \langle 5,9,-2\right \rangle$

Explanation:

In order to express 5i+9j-2k in vector form, we will need to map its , , and coefficients to its -, -, and -coordinates.

Thus, its vector form is

$\left \langle 5,9,-2\right \rangle$ .

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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1 : Vector Form

Example Question #2 : Vector Form

Example Question #25 : Linear Algebra

Example Question #2 : Vectors & Spaces

Example Question #1 : Vectors

Example Question #1 : Vector Form

Example Question #1 : Vector Form

Example Question #1 : Vector Form

Example Question #1 : Vector Form

Example Question #11 : Vector Form

All Calculus 2 Resources

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