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

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

Q: If $$\vec{u} = (1, 2)$$ and $$\vec{v} = (3, 4)$$, what is the angle $$\theta$$ between them? (Use $$\cos \theta = \dfrac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|})$$

$$\cos \theta = \dfrac{11}{\sqrt{5} \sqrt{25}}$$

Question 1

Which of the following statements best describes the magnitude of a vector $$\vec{v}$$ from point $$A to $$point $$B$$?

Accepted Answer

It is the length of the line segment from $$A to$$ $$B$$, denoted $$|\vec{v}|.$$

Question 2

Given vectors $$\vec{u} = 2\hat{i} + 3\hat{j}$$ and $$\vec{v} = -\hat{i} + 4\hat{j}$$, what is $$\vec{u} + \vec{v}$$?

Accepted Answer

$$\hat{i} + 7\hat{j}$$

Question 3

If $$\vec{u} = 3\hat{i} - 4\hat{j}$$, what is the magnitude $$|\vec{u}|$$?

Accepted Answer

$$5$$

Question 4

Which property of vector addition is illustrated by $$\vec{u} + (\vec{v} + \vec{w}) = (\vec{u} + \vec{v}) + \vec{w}$$?

Accepted Answer

Associative property

Question 5

A force $$\vec{F} = 6\hat{i} + 8\hat{j} N $$acts on an object. What is the magnitude of the force?

Accepted Answer

$$10 N$$

Question 6

If $$\vec{u} = 2\hat{i} + 4\hat{j} - 5\hat{k}$$ and $$\vec{v} = \hat{i} + 2\hat{j} + \hat{k}$$, what is $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

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

Question 7

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

Accepted Answer

$$\vec{v} \cdot \dfrac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2}$$

Question 8

If $$\vec{u} = (a, b, c)$$ and $$\vec{v} = (d, e, f)$$, what is the dot product $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

$$ad + be + cf$$

Question 9

A vector $$\vec{u} is $$perpendicular to $$\vec{v} if $$and only if:

Accepted Answer

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

Question 10

Given $$\vec{u} = 2t\hat{i} + 4\hat{j} - (10 + t)\hat{k}$$ and $$\vec{v} = \hat{i} + t\hat{j} + \hat{k}$$, for what value of $$t$$ are $$\vec{u}$$ and $$\vec{v}$$ perpendicular?

Accepted Answer

$$t = 2$$

Question 11

Which of the following is the unit vector in the direction of $$\vec{u} = -3\hat{i} + 4\hat{j}$$?

Accepted Answer

$$-\dfrac{3}{5}\hat{i} + \dfrac{4}{5}\hat{j}$$

Question 12

If $$\vec{u} = (u_1$, u_2$, u_3$)$$, what is the formula for its magnitude $$|\vec{u}|$$?

Accepted Answer

$$\sqrt{u_1$^2 + u_2$^2 + u_3$^2}$$

Question 13

A plane travels from city $$A(1,2,3) to $$city $$B(4,6,8). $$What is the displacement vector $$\vec{d}$$?

Accepted Answer

$$(3, 4, 5)$$

Question 14

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

Accepted Answer

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

Question 15

If $$\vec{u} = (1, 2)$$ and $$\vec{v} = (3, 4)$$, what is the angle $$	heta$$ between them? (Use $$\cos 	heta = \dfrac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|})$$

Accepted Answer

$$\cos 	heta = \dfrac{11}{\sqrt{5} \sqrt{25}}$$

Question 16

A vector $$\vec{w} = 3\hat{i} + \hat{j} is $$expressed as the sum $$\vec{u} + \vec{v}$$, where $$\vec{u} \parallel \hat{i} + \hat{j}$$ and $$\vec{v} \perp \hat{i} + \hat{j}. $$What is $$\vec{u}$$?

Accepted Answer

$$2\hat{i} + 2\hat{j}$$

Question 17

Which of the following best describes the zero vector $$\vec{0}$$?

Accepted Answer

A vector with zero magnitude and no specific direction.

Question 18

If $$\vec{u} = (u_1$, u_2$, ..., u_n)$$ and $$\vec{v} = (v_1$, v_2$, ..., v_n)$$, what is the general formula for their dot product?

Accepted Answer

$$u_1$$ $$v_1$$ + $$u_2$$ $$v_2$$ + \cdots + u_n v_n$$

Vectors: Definitions, Operations, and Applications in Calculus

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