Question 1

Which of the following is the correct formula for the distance between two vectors $$\vec{u} = (u_1$, u_2$)$$ and $$\vec{v} = (v_1$, v_2$) in$$ $$\mathbb{R}^2$$?

Accepted Answer

$$d(\vec{u}, \vec{v}) = \sqrt{(u_1$ - v_1$)^2$$ + ($$u_2$$ - $$v_2)^2$}$$

Question 2

Given the vector $$\vec{a} = (3, -4)$$, what is its magnitude (length)?

Accepted Answer

$$\sqrt{3^2$ + (-4)^2$} = 5$$

Question 3

Which of the following statements about the zero vector $$\vec{0} in$$ $$\mathbb{R}^n is $$TRUE?

Accepted Answer

The zero vector has all components equal to zero and its magnitude is $$0.$$

Question 4

If $$\vec{u} = (2, 1)$$ and $$\vec{v} = (4, 2)$$, which of the following is TRUE?

Accepted Answer

$$\vec{u}$$ and $$\vec{v}$$ are collinear because $$\vec{v} = 2\vec{u}.$$

Question 5

Which of the following vectors is a unit vector?

Accepted Answer

$$\left( \frac{3}{5}, \frac{4}{5} \right)$$

Question 6

Let $$\vec{a} = (2, -3)$$ and $$\vec{b} = (4, -6). $$Which statement is correct?

Accepted Answer

$$\vec{b} is a $$scalar multiple of $$\vec{a}$$, so they are collinear.

Question 7

If $$\vec{u} = (x, y) is a $$unit vector, which equation must be satisfied?

Accepted Answer

$$x^2$$ + $$y^2 = 1$

Question 8

Given $$\vec{u} = (1, 2, 3)$$ and $$\vec{v} = (-1, 0, 1)$$, what is the distance between $$\vec{u}$$ and $$\vec{v}$$?

Accepted Answer

$$\sqrt{(1 - (-1))^2$ + (2 - 0)^2$ + (3 - 1)^2$} = \sqrt{4 + 4 + 4} = 2\sqrt{3}$$

Question 9

Which of the following best describes a scalar in the context of vectors?

Accepted Answer

A scalar is a real number that can multiply a vector, changing its magnitude but not its direction (unless the scalar is negative).

Question 10

If $$\vec{u} = (a, b)$$ and $$\vec{v} = (ka, kb)$$ for some real number $$k 
eq 0$$, what is the relationship between $$\vec{u}$$ and $$\vec{v}$$?

Accepted Answer

$$\vec{u}$$ and $$\vec{v}$$ are collinear.

Question 11

Which of the following is the correct formula for the length (magnitude) of a vector $$\vec{v} = (v_1$, v_2$, v_3$) in$$ $$\mathbb{R}^3$$?

Accepted Answer

$$\|\vec{v}\| = \sqrt{v_1$^2 + v_2$^2 + v_3$^2}$$

Question 12

Suppose $$\vec{u} = (2, 3)$$ and $$\vec{v} = (4, 6). $$What is the unit vector in the same direction as $$\vec{v}$$?

Accepted Answer

$$\left( \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \right)$$

Question 13

If $$\vec{a} = (1, 2)$$ and $$\vec{b} = (2, 4)$$, and $$\vec{c} = (3, 6)$$, which statement is TRUE?

Accepted Answer

All three vectors are collinear.

Question 14

Which of the following is NOT a property of the zero vector $$\vec{0}$$?

Accepted Answer

It is a unit vector.

Question 15

If $$\vec{u} = (x, y)$$ and $$\vec{v} = (y, x)$$, for which values of $$x$$ and $$y$$ are $$\vec{u}$$ and $$\vec{v}$$ collinear (other than the zero vector)?

Accepted Answer

For $$x = y 
eq 0 or$$ $$x = -y 
eq 0$$

Question 16

Given $$\vec{u} = (1, 2, 3)$$, what is the length of $$-2\vec{u}$$?

Accepted Answer

$$2\sqrt{14}$$

Question 17

Which of the following best describes when two vectors in $$\mathbb{R}^n$$ are collinear?

Accepted Answer

They are scalar multiples of each other.

Question 18

If $$\vec{u} = (a, b) is a $$unit vector and $$a \u003e 0$$, $$b \u003e 0$$, which of the following could be $$\vec{u}$$?

Accepted Answer

$$\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)$$

Question 19

Suppose $$\vec{u} = (2, -1)$$ and $$\vec{v} = (k, -0.5)$$ are collinear. What is the value of $$k$$?

Accepted Answer

$$k = 4$$

Vectors: Definitions, Properties, and Applications (Math 1229A/B Unit 1)

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