Q: Given vectors $$\vec{a} = (2, -1, 3)$$ and $$\vec{b} = (4, 0, -2)$$, what is the value of their dot product $$\vec{a} \cdot \vec{b}$$?

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

Q: If $$\vec{a} = (1, 2, 3)$$ and $$\vec{b} = (4, 5, 6)$$, what is the angle $$\theta$$ between $$\vec{a}$$ and $$\vec{b}$$? (Use $$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|})$$

$$\theta = \cos$$^{-1}$$ \left( \frac{32}{\sqrt{14} \sqrt{77}} \right)$$

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

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

Q: If $$\vec{a} = (0, 1, 0)$$ and $$\vec{b} = (2, 3, 5)$$, what is the angle between $$\vec{a}$$ and $$\vec{b}$$? (Use $$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|})$$

$$\theta = \cos$$^{-1}$$ \left( \frac{3}{\sqrt{1} \sqrt{38}} \right)$$

Q: If $$\vec{a} = (2, 3, 1)$$ and $$\vec{b} = (1, 2, 3)$$, what is the value of $$5\vec{a} \cdot (\vec{a} - 2\vec{b})$$?

$$5 \times [\vec{a} \cdot \vec{a} - 2\vec{a} \cdot \vec{b}] = 5 \times [14 - 2 \times 11] = 5 \times [-8] = -40$$

Question 1

Given vectors $$\vec{a} = (2, -1, 3)$$ and $$\vec{b} = (4, 0, -2)$$, what is the value of their dot product $$\vec{a} \cdot \vec{b}$$?

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 $$\vec{a} = (1, 2, 3)$$ and $$\vec{b} = (4, 5, 6)$$, what is the angle $$	heta$$ between $$\vec{a}$$ and $$\vec{b}$$? (Use $$\cos 	heta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|})$$

Accepted Answer

$$	heta = \cos$$^{-1}$$ \left( \frac{32}{\sqrt{14} \sqrt{77}} ight)$$

Question 4

Which property of the dot product is illustrated by $$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$$?

Accepted Answer

Commutativity

Question 5

If $$\vec{a} = (3, 0, 1)$$ and $$\vec{b} = (0, -1, 0)$$, what is the angle between $$\vec{a}$$ and $$\vec{b}$$?

Accepted Answer

$$90^\circ$$

Question 6

A force $$\vec{F} = (10, 0, 0) N is $$applied to move an object 5 meters in the direction $$\vec{d} = (1, 0, 0). $$What is the work done by the force?

Accepted Answer

$$50 J$$

Question 7

If $$\vec{a} = (1, 2, 2)$$, what is the scalar projection of $$\vec{b} = (2, 1, 1)$$ onto $$\vec{a}$$?

Accepted Answer

$$\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|} = \frac{2 + 2 + 2}{\sqrt{1^2$ + 2^2$ + 2^2$}} = \frac{6}{3} = 2$$

Question 8

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

Accepted Answer

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

Question 9

If the angle between two nonzero vectors is $$90^\circ$$, what is their dot product?

Accepted Answer

$$0$$

Question 10

Given $$\vec{a} = (1, 1, 1)$$ and $$\vec{b} = (2, 2, 2)$$, what is the value of $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$6$$

Question 11

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 12

If $$\vec{a} = (0, 1, 0)$$ and $$\vec{b} = (2, 3, 5)$$, what is the angle between $$\vec{a}$$ and $$\vec{b}$$? (Use $$\cos 	heta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|})$$

Accepted Answer

$$	heta = \cos$$^{-1}$$ \left( \frac{3}{\sqrt{1} \sqrt{38}} ight)$$

Question 13

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

Accepted Answer

It is the product of the magnitudes of $$\vec{a}$$ and $$\vec{b}$$ and the cosine of the angle between them.

Question 14

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

Accepted Answer

$$\frac{32}{\sqrt{14}}$$

Question 15

A medical device applies a force $$\vec{F} = (3, 4, 0) N to $$move a surgical tool along a displacement $$\vec{d} = (2, 0, 0) m. $$What is the work done by the device?

Accepted Answer

$$6 J$$

Question 16

If $$\vec{a} = (1, 0, 0)$$ and $$\vec{b} = (0, 1, 0)$$, what is $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$0$$

Question 17

If $$\vec{a} = (2, 3, 1)$$ and $$\vec{b} = (1, 2, 3)$$, what is the value of $$5\vec{a} \cdot (\vec{a} - 2\vec{b})$$?

Accepted Answer

$$5 	imes [\vec{a} \cdot \vec{a} - 2\vec{a} \cdot \vec{b}] = 5 	imes [14 - 2 	imes 11] = 5 	imes [-8] = -40$$

Question 18

Which of the following real-world scenarios can be modeled using the dot product?

Accepted Answer

Calculating the work done by a force acting along a displacement.

Section 2.3 - The Dot Product

Guida di studio - Domande di pratica

Study guide - Flashcard