Q: Which of the following is the correct formula for the dot product of $$\vec{a}$$ and $$\vec{b} in $$terms of the angle $$\theta$$ between them?

$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta)$$

Question 1

Given vectors $$\vec{a} = \langle 2, 1 \rangle$$ and $$\vec{b} = \langle 3, -1 \rangle$$, what is the value of their dot product $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$5$$

Question 2

Which of the following statements about the dot product is TRUE?

Accepted Answer

The dot product of two vectors is a scalar.

Question 3

If $$\vec{u} = \langle 3, 1, 7 \rangle$$ and $$\vec{v} = \langle 1, 5, 2 \rangle$$, what is $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

$$22$$

Question 4

Which of the following is the correct formula for the dot product of $$\vec{a}$$ and $$\vec{b} in $$terms of the angle $$	heta$$ between them?

Accepted Answer

$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(	heta)$$

Question 5

If $$\vec{u} \cdot \vec{v} = 0$$ and neither $$\vec{u}$$ nor $$\vec{v} is $$the zero vector, what can be concluded about $$\vec{u}$$ and $$\vec{v}$$?

Accepted Answer

They are orthogonal.

Question 6

Given $$\vec{a} = \langle 4, -4, 1 \rangle$$ and $$\vec{b} = \langle 0, 4, -1 \rangle$$, what is $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$0$$

Question 7

Which property does the dot product satisfy for any scalar $$c$$ and vectors $$\vec{u}$$, $$\vec{v}$$?

Accepted Answer

$$c(\vec{u} \cdot \vec{v}) = (c\vec{u}) \cdot \vec{v} = \vec{u} \cdot (c\vec{v})$$

Question 8

If $$\vec{a} = \langle 1, 2, 3 \rangle$$, what is the projection of $$\vec{a}$$ onto the $$\vec{i}$$ direction?

Accepted Answer

$$\langle 1, 0, 0 \rangle$$

Question 9

Suppose $$\vec{u} = \langle 2, 1 angle$$ and $$\vec{v} = \langle 3, -1 angle. $$What is the angle $$	heta$$ between $$\vec{u}$$ and $$\vec{v}$$?

Accepted Answer

$$\dfrac{\pi}{4}$$

Question 10

Which of the following best describes the geometric meaning of the dot product $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

It measures how much of $$\vec{a}$$ goes in the direction of $$\vec{b}.$$

Question 11

Given $$\vec{a} = \langle 2, 1 \rangle$$ and $$\vec{b} = \langle 3, -1 \rangle$$, what is $$|\vec{a}|$$?

Accepted Answer

$$\sqrt{5}$$

Question 12

If $$\vec{u} = \langle 0, 1, 0 \rangle$$ and $$\vec{v} = \langle 0, 1, 0 \rangle$$, what is $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

$$1$$

Question 13

Which of the following is the correct formula for the orthogonal projection of $$\vec{w}$$ onto $$\vec{v}$$?

Accepted Answer

$$\mathrm{proj}_{\vec{v}} \vec{w} = \dfrac{\vec{w} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v}$$

Question 14

A force $$\vec{F} = \langle 10, 0 \rangle N is $$applied to move an object 5 meters in the direction $$\vec{d} = \langle 3, 4 \rangle. $$What is the work done by the force?

Accepted Answer

$$30 J$$

Question 15

If $$\vec{a} = \langle a, b, c \rangle$$, what is the projection of $$\vec{a}$$ onto $$\vec{j}$$?

Accepted Answer

$$\langle 0, b, 0 \rangle$$

Question 16

Which of the following is NOT a property of the dot product?

Accepted Answer

The dot product of two vectors is always positive.

Question 17

If $$\vec{u} = \langle 1, 2, 3 \rangle$$ and $$\vec{v} = \langle 4, -4, 1 \rangle$$, what is the scalar projection of $$\vec{u}$$ onto $$\vec{v}$$?

Accepted Answer

$$\dfrac{5}{\sqrt{33}}$$

Question 18

Which of the following best describes the result of projecting a vector $$\vec{a}$$ onto the $$\vec{i}$$, $$\vec{j}$$, and $$\vec{k}$$ axes?

Accepted Answer

It gives the decomposition of $$\vec{a}$$ into its coordinate directions.

Dot Product and Vector Projections in Calculus

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