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Calculus 3 : Dot Product

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Example Questions

Example Question #1 : Dot Product

Evaluate the dot product between $u=(e^t, t^2, \ln(t))$ , and $v=(2, \ln(t),3)$ .

Possible Answers:

$u\cdot v=2e^t+3\ln(t)$

$u\cdot v=2e^t+t^2\ln(t)+3\ln(t)$

$u\cdot v=0$

$u\cdot v=e^t+2\ln(t)$

$u\cdot v=2+t^2+3$

Correct answer:

$u\cdot v=2e^t+t^2\ln(t)+3\ln(t)$

Explanation:

All we need to do is multiply like components.

$u\cdot v=2e^t+t^2\ln(t)+3\ln(t)$

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Example Question #2 : Dot Product

Evaluate the dot product of u=(1,2,3) , and v=(9,-2,-10) .

Possible Answers:

$u \cdot v =-20$

$u \cdot v =25$

$u \cdot v =-25$

$u \cdot v =0$

Correct answer:

$u \cdot v =-25$

Explanation:

All we need to do is multiply the like components and add them together.

$u\cdot v=(1)(9)+(2)(-2)+(3)(-10)=9-4-30=-25$

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Example Question #1 : Dot Product

Find the dot product of the following vectors:

$\small \bold{v_1}=(1,0,-1)$

$\small \bold{v_2}=(1,1,1)$

Possible Answers:

$\small 1$

$\small -1$

$\small 0$

$\small 2$

Correct answer:

$\small 0$

Explanation:

To find the dot product between two vectors

$\small \small \small \bold{v}=(v_1,v_2,v_3)$

$\small \small \small \bold{w}=(w_1,w_2,w_3)$

we calculate

$\small \bold{v}\cdot\bold{w}=v_1w_1+v_2w_2+v_3w_3$

so for

$\small \bold{v_1}=(1,0,-1)$

$\small \bold{v_2}=(1,1,1)$

we have

$\small \small \small \small \small \bold{v_1}\cdot \bold{v_2}=1\cdot 1+0\cdot 1-1\cdot 1=1-1=0$

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Example Question #2 : Dot Product

What is the length of the vector

$\small \bold{v}=(3,4,5)$ ?

Possible Answers:

$\small 5\sqrt{2}$

$\small 5$

$\small 50$

$\small \small 2\sqrt{5}$

Correct answer:

$\small 5\sqrt{2}$

Explanation:

We can compute the length of a vector by taking the square root of the dot product of $\small \bold{v}$ and $\small \bold{v}$ , so the length of $\small \bold{v}=(3,4,5)$ is:

$\small |\bold{v}|=\sqrt{\bold{v}\cdot \bold{v}}=\sqrt{3^2+4^2+5^2}=\sqrt{9+25+25}$

$\small =\sqrt{50}=\sqrt{25\cdot 2}=5\sqrt{2}$

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Example Question #3 : Dot Product

Find the dot product of the following vectors:

$\small \small \bold{v_1}=(1,2,3)$

$\small \small \bold{v_2}=(1,4,5)$

Possible Answers:

$\small 20$

$\small \sqrt{24}$

$\small 24$

Correct answer:

$\small 24$

Explanation:

To find the dot product between two vectors

$\small \small \small \bold{v}=(v_1,v_2,v_3)$

$\small \small \small \bold{w}=(w_1,w_2,w_3)$

we calculate

$\small \bold{v}\cdot\bold{w}=v_1w_1+v_2w_2+v_3w_3$

so for

$\small \small \bold{v_1}=(1,2,3)$

$\small \small \bold{v_2}=(1,4,5)$

we have

$\small \small \small \small \small \small \bold{v_1}\cdot \bold{v_2}=1\cdot 1+2\cdot 4+3\cdot 5=1+8+15=24$

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Example Question #4 : Dot Product

What is the length of the vector

$\small \small \bold{v}=(1,1,\sqrt{2})$ ?

Possible Answers:

$\small 4$

$\small 2$

$\small \sqrt{2}$

Correct answer:

$\small 2$

Explanation:

We can compute the length of a vector by taking the square root of the dot product of $\small \bold{v}$ and $\small \bold{v}$ , so the length of $\small \small \bold{v}=(1,1,\sqrt{2})$ is:

$\small \small |\bold{v}|=\sqrt{\bold{v}\cdot \bold{v}}=\sqrt{1^2+1^2+\sqrt{2}^2}=\sqrt{1+1+2}$

$\small \small =\sqrt{4}=2$

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Example Question #5 : Dot Product

Which of the following cannot be used as a definition of the dot product of two real-valued vectors?

Possible Answers:

$\mathbf{a} \cdot \mathbf{b} = \sum_{i = 1}^{n}a_ib_i$

$\mathbf{a} \cdot \mathbf{b} = \sum_{i = 1}^{n}a_i+b_i$

$\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\|\cos{\theta}$ , where $\theta$ is the angle between $\mathbf{a}, \mathbf{b}$ .

They may all be used

Correct answer:

$\mathbf{a} \cdot \mathbf{b} = \sum_{i = 1}^{n}a_i+b_i$

Explanation:

$\mathbf{a} \cdot \mathbf{b} = \sum_{i = 1}^{n}a_i+b_i$ is not correct. This is saying effectively to add all the components of the two vectors together. The other two definitions are commonly used in computing angles between vectors and other objects, and can also be derived from each other.

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Example Question #5 : Dot Product

Which of the following is true concerning the dot product of two vectors?

Possible Answers:

$\mathbf{a}\cdot\mathbf{b}\cdot\mathbf{c}$ is well-defined as long as each vector is the same dimension

The dot product of two vectors is never negative.

None of the other statements are true.

$\mathbf{a} \cdot \mathbf{b} =0$ if and only if $\mathbf{a}, \mathbf{b}$ are orthogonal.

The dot product of two vectors is never a scalar.

Correct answer:

$\mathbf{a} \cdot \mathbf{b} =0$ if and only if $\mathbf{a}, \mathbf{b}$ are orthogonal.

Explanation:

This statement is true; it can be derived from the definition $\mathbf{a}\cdot\mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta$ by setting the acute angle between the vectors to be $\theta = \frac{\pi}{2}$ ; the requirement for orthogonality. Additionally, if either vector has length , the vectors are still said to be orthogonal.

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Example Question #9 : Dot Product

What is the dot product of vectors $\vec{a}= -\hat{i}-\hat{j} -3\hat{k}$ and $\vec{b}= -3\hat{i}-4\hat{j} -5\hat{k}$ ?

Possible Answers:

Correct answer:

Explanation:

Let vector be represented as $\vec{a}= a_{1}\hat{i}+a_{2}\hat{j} +a_{3}\hat{k}$ and vector be represented as $\vec{b}= b_{1}\hat{i}+b_{2}\hat{j} +b_{3}\hat{k}$ .

The dot product of the vectors and is $a\cdot b=a_{1}( b_{1})+a_{2}( b_{2})+a_{3}( b_{3})$ .

In this problem

$a\cdot b=-1(-3)-1(-4)-3(-5)=22$

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Example Question #10 : Dot Product

What is the dot product of vectors $\vec{a}= 2\hat{i}+5\hat{j} +7\hat{k}$ and $\vec{b}= -2\hat{i}-3\hat{j} +\hat{k}$ ?

Possible Answers:

Does not exist

Correct answer:

Explanation:

Let vector be represented as $\vec{a}= a_{1}\hat{i}+a_{2}\hat{j} +a_{3}\hat{k}$ and vector be represented as $\vec{b}= b_{1}\hat{i}+b_{2}\hat{j} +b_{3}\hat{k}$ .

The dot product of the vectors and is $a\cdot b=a_{1}( b_{1})+a_{2}( b_{2})+a_{3}( b_{3})$ .

In this problem

$a\cdot b=2(-2)+5(-3)+7(1)=-12$

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Calculus 3 : Dot Product

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Dot Product

Example Question #2 : Dot Product

Example Question #1 : Dot Product

Example Question #2 : Dot Product

Example Question #3 : Dot Product

Example Question #4 : Dot Product

Example Question #5 : Dot Product

Example Question #5 : Dot Product

Example Question #9 : Dot Product

Example Question #10 : Dot Product

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