Q: Given vectors $$\mathbf{a} = \langle 3, 4 \rangle$$ and $$\mathbf{b} = \langle 1, -2 \rangle$$, what is the value of $$\mathbf{a} \cdot \mathbf{b}$$?

$$3 \times 1 + 4 \times (-2) = 3 - 8 = -5$$

Q: Which formula correctly gives the angle $$\theta$$ between two nonzero vectors $$\mathbf{a}$$ and $$\mathbf{b}$$?

$$\theta = \arccos \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} \right)$$

Q: A force $$\mathbf{F} = \langle 10, 0 \rangle N $$acts on an object that moves through a displacement $$\mathbf{d} = \langle 3, 4 \rangle m. $$What is the work done by the force?

$$10 \times 3 + 0 \times 4 = 30 J$$

Q: If $$\mathbf{a} = \langle 2, 3 \rangle$$ and $$\mathbf{b} = \langle 4, -1 \rangle$$, what is the angle $$\theta$$ between them in degrees? (Round to two decimal places.)

$$\theta = \arccos \left( \frac{2 \times 4 + 3 \times (-1)}{\sqrt{2^2$ + 3^2$} \sqrt{4^2$ + (-1)^2$}} \right ) = \arccos \left( \frac{8 - 3}{\sqrt{13} \sqrt{17}} \right ) = \arccos \left( \frac{5}{\sqrt{221}} \right ) \approx 70.35^\circ$$

Q: A triangle has sides $$OA = 5$$ units, $$OB = 3$$ units, and the angle between them is $$30^\circ. $$What is the length of side $$AB$$?

$$AB = \sqrt{5^2$ + 3^2$ - 2 \times 5 \times 3 \cos 30^\circ} \approx 2.83$$ units

Question 1

Given vectors $$\mathbf{a} = \langle 3, 4 \rangle$$ and $$\mathbf{b} = \langle 1, -2 \rangle$$, what is the value of $$\mathbf{a} \cdot \mathbf{b}$$?

Accepted Answer

$$3 	imes 1 + 4 	imes (-2) = 3 - 8 = -5$$

Question 2

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

Accepted Answer

The dot product is commutative: $$\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$$

Question 3

If $$\mathbf{a} = \langle 2, 5 \rangle$$ and $$\mathbf{b} = \langle x, 1 \rangle$$ are perpendicular, what is the value of $$x$$?

Accepted Answer

$$x = -\frac{5}{2}$$

Question 4

Which formula correctly gives the angle $$	heta$$ between two nonzero vectors $$\mathbf{a}$$ and $$\mathbf{b}$$?

Accepted Answer

$$	heta = \arccos \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} ight)$$

Question 5

A force $$\mathbf{F} = \langle 10, 0 \rangle N $$acts on an object that moves through a displacement $$\mathbf{d} = \langle 3, 4 \rangle m. $$What is the work done by the force?

Accepted Answer

$$10 	imes 3 + 0 	imes 4 = 30 J$$

Question 6

If $$\mathbf{a} = \langle 1, 2 \rangle$$, $$\mathbf{b} = \langle 3, -1 \rangle$$, and $$\mathbf{c} = \langle 2, 5 \rangle$$, what is $$\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})$$?

Accepted Answer

$$\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} = (1 	imes 3 + 2 	imes -1) + (1 	imes 2 + 2 	imes 5) = (3 - 2) + (2 + 10) = 1 + 12 = 13$$

Question 7

Given $$\mathbf{a} = \langle 4, 3 \rangle$$, what is the magnitude $$\|\mathbf{a}\|$$?

Accepted Answer

$$\sqrt{4^2$ + 3^2$} = \sqrt{16 + 9} = \sqrt{25} = 5$$

Question 8

If $$\mathbf{a} = \langle 2, 3 angle$$ and $$\mathbf{b} = \langle 4, -1 angle$$, what is the angle $$	heta$$ between them in degrees? (Round to two decimal places.)

Accepted Answer

$$	heta = \arccos \left( \frac{2 	imes 4 + 3 	imes (-1)}{\sqrt{2^2$ + 3^2$} \sqrt{4^2$ + (-1)^2$}} ight ) = \arccos \left( \frac{8 - 3}{\sqrt{13} \sqrt{17}} ight ) = \arccos \left( \frac{5}{\sqrt{221}} ight ) \approx 70.35^\circ$$

Question 9

Which of the following is the correct expression for the square of the magnitude of $$\mathbf{a} - \mathbf{b} in $$terms of $$\mathbf{a}$$, $$\mathbf{b}$$, and the angle $$	heta$$ between them?

Accepted Answer

$$\|\mathbf{a} - \mathbf{b}\|^2 = \|\mathbf{a}\|^2 + \|\mathbf{b}\|^2 - 2\|\mathbf{a}\|\|\mathbf{b}\| \cos 	heta$$

Question 10

A triangle has sides $$OA = 5$$ units, $$OB = 3$$ units, and the angle between them is $$30^\circ. $$What is the length of side $$AB$$?

Accepted Answer

$$AB = \sqrt{5^2$ + 3^2$ - 2 	imes 5 	imes 3 \cos 30^\circ} \approx 2.83$$ units

Question 11

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

Accepted Answer

The dot product of two perpendicular nonzero vectors is always nonzero.

Question 12

If $$\mathbf{a} = \langle 1, 2 \rangle$$ and $$\mathbf{b} = \langle 9, -7 \rangle$$, what is $$\mathbf{a} \cdot (2\mathbf{b})$$?

Accepted Answer

$$\mathbf{a} \cdot (2\mathbf{b}) = 1 	imes 18 + 2 	imes (-14) = 18 - 28 = -10$$

Question 13

If $$\mathbf{a} = \langle a_1$, a_2$ \rangle$$, what is $$\mathbf{a} \cdot \mathbf{a}$$?

Accepted Answer

$$a_1^2$$ + $$a_2^2$

Question 14

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

Accepted Answer

It equals $$\|\mathbf{a}\| \|\mathbf{b}\| \cos 	heta$$, where $$	heta is $$the angle between $$\mathbf{a}$$ and $$\mathbf{b}.$$

Question 15

If $$\mathbf{a} = \langle 1, 2 \rangle$$ and $$\mathbf{b} = \langle 3, x \rangle$$ are perpendicular, what is the value of $$x$$?

Accepted Answer

$$x = -\frac{3}{2}$$

Question 16

Which of the following is a correct statement about the distributive property of the dot product?

Accepted Answer

$$\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$$

Question 17

If $$\mathbf{a} = \langle 5, 1 \rangle$$ and $$\mathbf{b} = \langle 2, 4 \rangle$$, what is the angle between them in radians? (Round to two decimal places.)

Accepted Answer

$$	heta = \arccos \left( \frac{5 	imes 2 + 1 	imes 4}{\sqrt{5^2$ + 1^2$} \sqrt{2^2$ + 4^2$}} ight ) = \arccos \left( \frac{14}{\sqrt{26} \sqrt{20}} ight ) \approx 0.91$$ radians

Mechanics I: The Dot Product and Its Geometrical Applications

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