Q: Given two vectors $$\vec{u} = (2, -1, 3)$$ and $$\vec{v} = (4, 0, -2)$$, what is their dot product $$\vec{u} \cdot \vec{v}$$?

$$2 \times 4 + (-1) \times 0 + 3 \times (-2) = 8 + 0 - 6 = 2$$

Q: If $$\vec{a} = (1, 2, 2)$$ and $$\vec{b} = (2, 1, -2)$$, what is the angle $$\theta$$ between $$\vec{a}$$ and $$\vec{b}$$? (Use $$\cos \theta = \dfrac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|})$$

$$\cos \theta = \dfrac{1 \times 2 + 2 \times 1 + 2 \times (-2)}{\sqrt{1^2$ + 2^2$ + 2^2$} \sqrt{2^2$ + 1^2$ + (-2)^2$}} = \dfrac{2 + 2 - 4}{\sqrt{9} \sqrt{9}} = 0$$

Q: Suppose a force $$\vec{F} = (10, 0, 0) N is $$applied to move an object 5 meters along the vector $$\vec{d} = (3, 4, 0)$$ meters. What is the work done by the force?

$$W = \vec{F} \cdot \vec{d} = 10 \times 3 + 0 \times 4 + 0 \times 0 = 30$$ Joules

Q: A vector $$\vec{a}$$ has magnitude $$5$$ and a vector $$\vec{b}$$ has magnitude $$12. If $$the angle between them is $$90^\circ$$, what is $$\vec{a} \cdot \vec{b}$$?

$$\vec{a} \cdot \vec{b} = 5 \times 12 \times \cos 90^\circ = 0$$

Q: If $$\vec{u} = (1, 2, 3)$$ and $$\vec{v} = (4, 5, 6)$$, what is the projection of $$\vec{u}$$ onto $$\vec{v}$$?

$$\text{proj}_{\vec{v}} \vec{u} = \dfrac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|^2} \vec{v} = \dfrac{32}{77} (4, 5, 6)$$

Question 1

Given two vectors $$\vec{u} = (2, -1, 3)$$ and $$\vec{v} = (4, 0, -2)$$, what is their dot product $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

$$2 	imes 4 + (-1) 	imes 0 + 3 	imes (-2) = 8 + 0 - 6 = 2$$

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 $$\vec{a} = (1, 2, 2)$$ and $$\vec{b} = (2, 1, -2)$$, what is the angle $$	heta$$ between $$\vec{a}$$ and $$\vec{b}$$? (Use $$\cos 	heta = \dfrac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|})$$

Accepted Answer

$$\cos 	heta = \dfrac{1 	imes 2 + 2 	imes 1 + 2 	imes (-2)}{\sqrt{1^2$ + 2^2$ + 2^2$} \sqrt{2^2$ + 1^2$ + (-2)^2$}} = \dfrac{2 + 2 - 4}{\sqrt{9} \sqrt{9}} = 0$$

Question 4

Which of the following is a necessary and sufficient condition for two nonzero vectors $$\vec{u}$$ and $$\vec{v} to be $$perpendicular?

Accepted Answer

$$\vec{u} \cdot \vec{v} = 0$$

Question 5

Suppose a force $$\vec{F} = (10, 0, 0) N is $$applied to move an object 5 meters along the vector $$\vec{d} = (3, 4, 0)$$ meters. What is the work done by the force?

Accepted Answer

$$W = \vec{F} \cdot \vec{d} = 10 	imes 3 + 0 	imes 4 + 0 	imes 0 = 30$$ Joules

Question 6

If $$\vec{u} = (a, b, c)$$ and $$\vec{v} = (d, e, f)$$, which formula correctly computes their dot product?

Accepted Answer

$$\vec{u} \cdot \vec{v} = a d + b e + c f$$

Question 7

A vector $$\vec{a}$$ has magnitude $$5$$ and a vector $$\vec{b}$$ has magnitude $$12. If $$the angle between them is $$90^\circ$$, what is $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$\vec{a} \cdot \vec{b} = 5 	imes 12 	imes \cos 90^\circ = 0$$

Question 8

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

Accepted Answer

It equals the product of the magnitude of $$\vec{u}$$, the magnitude of $$\vec{v}$$, and the cosine of the angle between them.

Question 9

If $$\vec{u} = (1, 2, 3)$$ and $$\vec{v} = (4, 5, 6)$$, what is the projection of $$\vec{u}$$ onto $$\vec{v}$$?

Accepted Answer

$$	ext{proj}_{\vec{v}} \vec{u} = \dfrac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|^2} \vec{v} = \dfrac{32}{77} (4, 5, 6)$$

Question 10

A plane in 3D space is defined by the equation $$2x - 3y + 4z = 12. $$What is a normal vector to this plane?

Accepted Answer

$$(2, -3, 4)$$

Question 11

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

Accepted Answer

$$\vec{u} \cdot \vec{u} = 0$$ for any vector $$\vec{u}$$

Question 12

A vector $$\vec{a} is $$parallel to vector $$\vec{b}. $$Which of the following is always true?

Accepted Answer

$$\vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\|$$

Question 13

In physics, the work done by a constant force $$\vec{F}$$ moving an object along a displacement $$\vec{d} is $$given by $$W = \vec{F} \cdot \vec{d}. If$$ $$\vec{F}$$ and $$\vec{d}$$ are perpendicular, what is the work done?

Accepted Answer

$$0$$

Question 14

If $$\vec{u} = (3, 4)$$ and $$\vec{v} = (4, -3)$$, what is $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

$$3 	imes 4 + 4 	imes (-3) = 12 - 12 = 0$$

Question 15

Which of the following real-world scenarios best illustrates the use of the dot product?

Accepted Answer

Calculating the work done by a force acting at an angle to the direction of motion.

Question 16

If $$\vec{u} = (1, 2, 2)$$, what is $$\vec{u} \cdot \vec{u}$$?

Accepted Answer

$$1^2$$ + $$2^2$$ + $$2^2 = 1 + 4 + 4 = 9$

Question 17

Given $$\vec{u} = (x_1$, y_1$, z_1$)$$ and $$\vec{v} = (x_2$, y_2$, z_2$)$$, which of the following expressions gives the cosine of the angle $$	heta$$ between them?

Accepted Answer

$$\cos 	heta = \dfrac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|}$$

The Dot Product: Concepts and Applications in Vector Calculus

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