Using Angles With Vectors

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education

Vectors are entities that have a magnitude and a direction associated with them. In this lesson, we will investigate how to use this information to determine rectangular coordinates and polar coordinates of vectors.

Vectors

Every car has a speedometer, which tells the driver how fast they are moving. This is a speed, but not a velocity. To be a velocity, the instrument panel would have to indicate the direction the car is moving. Velocity is a vector, which includes a magnitude (speed) and a direction. For example, a car is moving at 60 miles-per-hour, at 70°.

Rectangular Coordinates

Vectors are represented by arrows. Often times it is helpful to know where a vector ends by listing that location's rectangular coordinates. Rectangular coordinates are represented as (x,y). Let's look at an example where we can determine the termination point of a vector.

Example 1

Prompt: A car has a velocity of 60 miles-per-hour, at 70°. This is a polar coordinate because it gives a length of a vector and its angle relative to the x-axis. Represent this vector on a grid and determine its termination point in rectangular coordinates.

Solution: First, we represent this vector on an x-y grid.


The velocity vector is represented by the red arrow
ex1


Notice in the diagram of the vector there is a component of the velocity in the x-direction (eastward) and a component of the velocity in the y-direction (northward). To determine these component velocities, we have to use trigonometric functions.


Trigonometric functions
trig


We'll use the sine function to determine the northward component of the car's velocity and the cosine function to determine the eastward component of the car's velocity.


Determining the x and y-components of a velocity vector
ex1more


Starting with the northward velocity we get


Calculating the northward velocity
vy1


Now we will use the cosine function to determine the eastward component of the car's velocity.


Calculating the eastward velocity
vx1


The rectangular coordinates are (46.4,38). Let's now work an example where we are given rectangular coordinates and we have to turn them into polar coordinates.

Polar Coordinates

Polar coordinates represent the same thing as rectangular coordinates, but the information is given in a different manner. It is like saying something in the English language and then saying the same thing in the French language. Let's work an example changing rectangular coordinates into polar coordinates.

Example 2

Prompt: A bicyclist is traveling 5 m/s east and 1 m/s north (5,1). Determine the velocity of the bicyclist in polar coordinates.

The point at the tip of the red vector is the location (5,1)
ex2

Solution: We can use the Pythagoran theorem to determine the magnitude of vector v or we can use the tangent function. Let's use the tangent function.


Calculating the angle of the vector
polar1


We can use either the sine or cosine functions to determine the magnitude of vector, v. Let's use cosine.


Calculating the magnitude of the velocity
polar2


The velocity of the bicyclist approximately 5.1 m/s, at 11°.

Now let's learn how to add and subtract vectors.

Adding and Subtracting Vectors

We can compare adding and subtracting vectors to adding and subtracting variables in algebra. For example. 2x + y + 3x can be simplified to give 5x + y. We are allowed to add the x-terms because they have the same variable. The same rule applies to vectors except we are talking about component vectors. We can only add x-components to x-components, and y-components to y-components. We use unit vectors to represent pure directions. î represents x-direction vectors, and ĵ represents y-direction vectors.

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