Question 1

Given a vector $$\vec{v} = 4\hat{i} + 3\hat{j}$$, what is the magnitude of $$\vec{v}$$?

Accepted Answer

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

Question 2

What is the unit vector in the direction of $$\vec{v} = 4\hat{i} + 3\hat{j}$$?

Accepted Answer

$$\frac{4}{5}\hat{i} + \frac{3}{5}\hat{j}$$

Question 3

If $$\vec{a} = 2\hat{i} + 5\hat{j}$$, what is the unit vector in the direction of $$\vec{a}$$?

Accepted Answer

$$\frac{2}{\sqrt{29}}\hat{i} + \frac{5}{\sqrt{29}}\hat{j}$$

Question 4

A force of $$10 N $$acts in the direction of $$2\hat{i} + 3\hat{j}. $$What is the vector representation of this force?

Accepted Answer

$$\frac{10}{\sqrt{13}}(2\hat{i} + 3\hat{j})$$

Question 5

Which of the following best describes a unit vector?

Accepted Answer

A vector with magnitude $$1$$ and a specified direction.

Question 6

If $$\vec{b} = 6\hat{i} - 8\hat{j}$$, what is the unit vector in the direction of $$\vec{b}$$?

Accepted Answer

$$\frac{6}{10}\hat{i} - \frac{8}{10}\hat{j}$$

Question 7

A vector $$\vec{c}$$ has a magnitude of $$7$$ and points in the direction of $$\hat{i} + 2\hat{j}. $$What is $$\vec{c} in $$component form?

Accepted Answer

$$\frac{7}{\sqrt{5}}\hat{i} + \frac{14}{\sqrt{5}}\hat{j}$$

Question 8

Which of the following is NOT a property of unit vectors?

Accepted Answer

Their magnitude is always greater than $$1.$$

Question 9

If $$\vec{d} = a\hat{i} + b\hat{j} is a $$unit vector, which equation must be satisfied?

Accepted Answer

$$a^2$$ + $$b^2 = 1$

Question 10

A plane is flying in the direction $$3\hat{i} + 4\hat{j}. If $$its speed is $$100 km/h$$, what is its velocity vector?

Accepted Answer

$$60\hat{i} + 80\hat{j}$$

Question 11

If $$\vec{u} is a $$unit vector in the direction of $$\vec{v}$$, which of the following is true?

Accepted Answer

$$\vec{u} = \frac{\vec{v}}{\|\vec{v}\|}$$

Question 12

A vector $$\vec{w}$$ has components $$x$$ and $$y. If$$ $$\vec{w} is a $$unit vector, what is the value of $$x if$$ $$y = \frac{\sqrt{3}}{2}$$?

Accepted Answer

$$\frac{1}{2}$$

Unit Vectors and Direction in Trigonometry

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