Q: Which formula is used to find the angle $$\theta$$ between two vectors $$\vec{a}$$ and $$\vec{b}$$ using their dot product?

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

Q: If $$\vec{a} = (1, 2)$$ and $$\vec{b} = (2, 1)$$, which of the following is the correct calculation for $$\vec{a} \cdot \vec{b}$$?

$$1 \times 2 + 2 \times 1 = 4$$

Q: A vector $$\vec{a}$$ has magnitude $$5$$ and a vector $$\vec{b}$$ has magnitude $$12. If $$the angle between them is $$60^\circ$$, what is $$\vec{a} \cdot \vec{b}$$?

$$5 \times 12 \times \cos 60^\circ = 30$$

Q: If $$\vec{a} = (2, 3)$$ and $$\vec{b} = (4, 5)$$, what is the cosine of the angle between $$\vec{a}$$ and $$\vec{b}$$?

$$\dfrac{2 \times 4 + 3 \times 5}{\sqrt{2^2$ + 3^2$} \sqrt{4^2$ + 5^2$}}$$

Q: A vector $$\vec{a}$$ has magnitude $$8$$ and a vector $$\vec{b}$$ has magnitude $$15. If$$ $$\vec{a} \cdot \vec{b} = 60$$, what is the angle between them?

$$\cos$$^{-1}$$\left(\dfrac{60}{8 \times 15}\right)$$

Question 1

Which of the following correctly expresses the dot product of vectors $$\vec{a} = (a_1$, a_2$)$$ and $$\vec{b} = (b_1$, b_2$)$$ using components?

Accepted Answer

$$\vec{a} \cdot \vec{b} = a_1$ b_1$ + a_2$ b_2$

Question 2

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

Accepted Answer

$$0$$

Question 3

The dot product of two vectors is zero. What does this imply about the vectors?

Accepted Answer

They are perpendicular.

Question 4

Given $$\vec{a} = (2, 5)$$ and $$\vec{b} = (4, 1)$$, calculate $$\vec{a} \cdot \vec{b}.$$

Accepted Answer

$$13$$

Question 5

Which formula is used to find the angle $$	heta$$ between two vectors $$\vec{a}$$ and $$\vec{b}$$ using their dot product?

Accepted Answer

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

Question 6

If $$\vec{a} = (6, 8)$$ and $$\vec{b} = (3, 4)$$, what is the angle between $$\vec{a}$$ and $$\vec{b}$$?

Accepted Answer

$$0^\circ$$

Question 7

A force $$\vec{F} = (10, 0) N is $$applied to move an object along a displacement $$\vec{d} = (5, 0) m. $$What is the work done by the force?

Accepted Answer

$$50 J$$

Question 8

If $$\vec{a} = (1, 2)$$ and $$\vec{b} = (2, 1)$$, which of the following is the correct calculation for $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$1 	imes 2 + 2 	imes 1 = 4$$

Question 9

Given $$\vec{a} = (7, -2)$$ and $$\vec{b} = (-3, 4)$$, what is $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$-29$$

Question 10

If $$\vec{a} = (0, 5)$$ and $$\vec{b} = (3, 0)$$, what is the value of $$\vec{a} \cdot \vec{b}$$ and what does it indicate about the vectors?

Accepted Answer

$$0$$, the vectors are perpendicular.

Question 11

A vector $$\vec{a}$$ has magnitude $$5$$ and a vector $$\vec{b}$$ has magnitude $$12. If $$the angle between them is $$60^\circ$$, what is $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$5 	imes 12 	imes \cos 60^\circ = 30$$

Question 12

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

Accepted Answer

The dot product of two perpendicular vectors is nonzero.

Question 13

If $$\vec{a} = (2, 3)$$ and $$\vec{b} = (4, 5)$$, what is the cosine of the angle between $$\vec{a}$$ and $$\vec{b}$$?

Accepted Answer

$$\dfrac{2 	imes 4 + 3 	imes 5}{\sqrt{2^2$ + 3^2$} \sqrt{4^2$ + 5^2$}}$$

Question 14

A vector $$\vec{a} is $$perpendicular to vector $$\vec{b}. $$Which of the following must be true?

Accepted Answer

$$\vec{a} \cdot \vec{b} = 0$$

Question 15

If $$\vec{a} = (1, 2)$$ and $$\vec{b} = (2, 4)$$, what is the angle between $$\vec{a}$$ and $$\vec{b}$$?

Accepted Answer

$$0^\circ$$

Question 16

A vector $$\vec{a}$$ has magnitude $$8$$ and a vector $$\vec{b}$$ has magnitude $$15. If$$ $$\vec{a} \cdot \vec{b} = 60$$, what is the angle between them?

Accepted Answer

$$\cos$$^{-1}$$\left(\dfrac{60}{8 	imes 15}ight)$$

Question 17

If $$\vec{a} = (5, 12)$$ and $$\vec{b} = (12, 5)$$, what is $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$119$$

Question 18

A vector $$\vec{a}$$ has magnitude $$10$$ and a vector $$\vec{b}$$ has magnitude $$10. If $$the angle between them is $$90^\circ$$, what is $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$0$$

Question 19

Which of the following is a real-world application of the dot product?

Accepted Answer

Finding the work done by a force along a displacement.

Dot Product: Concepts, Calculations, and Applications

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