While working in your math class at school, the instructor passes everyone a map of Your town. She asks you to find your house and place a red dot on it, and then find the school and place a blue dot there. Your map looks like this:

She then asks you to break down the trip to your school in terms of component vectors and unit vectors. Are you able to do this?

### Unit Vectors and Components

A **unit vector** is a vector that has a magnitude of one unit and can have any direction. Traditionally

We are not studying 3D space in this course. The unit vector notation may seem burdensome but one must distinguish between a vector and the components of that vector in the direction of the

**Component vectors** of a given vector are two or more vectors whose sum is the given vector. The sum is viewed as equivalent to the original vector. Since component vectors can have any direction, it is useful to have them perpendicular to one another. Commonly one chooses the

A vector from the origin (0, 0) to the point (8, 0) is written as

The reason for having the component vectors perpendicular to one another is that this condition allows us to use the Pythagorean Theorem and trigonometric ratios to find the magnitude and direction of the components. One can solve vector problems without use of unit vectors if specific information about orientation or direction in space such as N, E, S or W is part of the problem.

Let's take a look at a few example problems.

1. What are the component vectors of the vector shown here?

Since the length of the vector is 5, and the angle the vector makes with the

And the "y" component is:

And we have the familiar 3, 4, 5 triangle, where the vector is the hypotenuse.

2. Why are unit vectors required when dealing with vector addition?

Unit vectors are required because it is necessary to have like quantities for addition. If there are two numbers, they can be added. If there are two vectors, they can be added. But if you have a number and a vector, they can't be added. Having unit vectors along with a magnitude makes a quantity a vector.

3. What are the unit vectors and the lengths of the component vectors when

The unit vectors in this case are

The length of the component vector in the

### Examples

#### Example 1

Earlier, you were asked to break down the trip to your school in terms of component vectors and unit vectors.

In this section, you learned that breaking a vector down into its components involves adding the portion of the vector along the "y" axis to the portion of the vector along the "x" axis. To accomplish this in the case of the map, you only need to write down the length the vector has in the "x" direction (along with an "x" unit vector) and then add to it the length the vector has in the "y" direction (along with a "y" unit vector). Your map should look like this:

#### Example 2

An inclined ramp is 12 feet long and forms an angle of

#### Example 3

A wind vector has a magnitude of 25 miles per hour with an angle of

Since the vector has an angle of

#### Example 4

A vector

The "x" component is

### Review

- Describe how to find the vertical and horizontal components of a vector when given the magnitude and direction of the vector.
a⃗ has a magnitude of 6 and a direction of100∘ . Find the components of the vector.b⃗ has a magnitude of 3 and a direction of60∘ . Find the components of the vector.c⃗ has a magnitude of 2 and a direction of84∘ . Find the components of the vector.d⃗ has a magnitude of 5 and a direction of32∘ . Find the components of the vector.- \begin{align*}\vec{e}\end{align*} has a magnitude of 2 and a direction of \begin{align*}45^\circ\end{align*}. Find the components of the vector.
- \begin{align*}\vec{f}\end{align*} has a magnitude of 7 and a direction of \begin{align*}70^\circ\end{align*}. Find the components of the vector.
- A plane is flying on a bearing of \begin{align*}50^\circ\end{align*} at 450 mph. Find the component form of the velocity of the plane. What does the component form tell you?
- A baseball is thrown at a \begin{align*}20^\circ\end{align*} angle with the horizontal with an initial speed of 30 mph. Find the component form of the initial velocity.
- A plane is flying on a bearing of \begin{align*}300^\circ\end{align*} at 500 mph. Find the component form of the velocity of the plane.
- A plane is flying on a bearing of \begin{align*}150^\circ\end{align*} at 470 mph. At the same time, there is a wind blowing at a bearing of \begin{align*}200^\circ\end{align*} at 60 mph. What is the component form of the velocity of the plane?
- Using the information from the previous problem, find the actual ground speed of the plane.
- Wind is blowing at a magnitude of 50 mph with an angle of \begin{align*}25^\circ\end{align*} with respect to the east. What is the velocity of the wind blowing to the north? What is the velocity of the wind blowing to the east?
- Find a unit vector in the direction of \begin{align*}\vec{a}\end{align*}, a vector in standard position with terminal point (-4, 3).
- Find a unit vector in the direction of \begin{align*}\vec{b}\end{align*}, a vector in standard position with terminal point (5, 1).

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.19.