Using Trigonometry to Work With Vectors

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Lots of physical quantities, like force, displacement, and velocity, are vectors. In this lesson, learn how to use trigonometry to work with vectors and get a chance to practice adding various kinds of vectors.

What is a Vector?

Amy wants to push her refrigerator across the floor so she gets a ladder, climbs it, and then pushes really hard on the top of the refrigerator. Do you think that will work? It's pretty clear that pushing on the top of the refrigerator won't make it move across the room!

What if instead, she applied the same amount of force to the side of the refrigerator? That's a much better idea, right? It takes force to move a refrigerator, but it's not enough to just know the amount of force you must apply, you also need to know what direction to apply the force. This is because a force is a type of mathematical quantity known as a vector. A vector is any quantity, such as force, that has both a magnitude (amount) and a direction. Some common examples of vector quantities are force, displacement, velocity, and acceleration.

What if you have several vectors and need to add them together? The sum of two vectors is known as the resultant, and you can use trigonometry to help you find it.

What to Do When the Vectors Form a Right Angle

Vectors can be represented graphically using an arrow. The length of the arrow should correspond to the magnitude of the vector, and the direction the arrow is drawn should correspond to the direction of the vector.

To add two vectors graphically, first draw one of the vectors on a piece of paper. Then draw the second vector from the END of the first vector. Make sure to draw arrows on the ends of the vectors so you know which direction the vectors are going.

For example, let's look at the following problem and see how to draw the vectors graphically:

  • If Sam leaves his house and walks 50 m in a straight line, then makes a 90-degree turn and walks another 20 m, how far from his house does he end up?

To begin to answer this question, draw the first displacement (50 m) and then the second displacement (20 m) from the end of the first, like this:


vector addition example 1_step 1


To find the resultant, which will tell you how far Sam is from his house, draw a vector from the BEGINNING of the first vector to the END of the second vector. This makes a triangle, so you can use some trigonometry to find both the magnitude and direction of the resultant.

There are two basic ways that you can use trigonometry to find the resultant of two vectors, and which method you need depends on whether or not the vectors form a right angle or not. In this case, they do form a right angle, so you can use the Pythagorean Theorem to find the length of the hypotenuse of the triangle, which will give you the magnitude of the resultant.


vector_addition_example_1_step_2


Remember that all vectors have both a magnitude and a direction. That's what makes them vectors! You just found the magnitude of the resultant, but you also need to find its direction. To find it, you can use the trigonometric function tangent, which is equal to the side opposite the angle (20 m) divided by the side adjacent to the angle (50 m).


vector addition example 1_step 3


So, in this case, Sam ends up a distance of 53.9 m from where he started along a line that is 22 degrees from his original direction. You now have both the magnitude and the direction of the resultant!

What to Do When the Vectors DO NOT Form a Right Angle

How is this process different when the vectors do not form a right angle? Let's look at an example.

  • Two forces are applied to the same object. A 10 N force is applied horizontally, and a 25 N force is applied at an angle of 30 degrees to the horizontal. What is the total force applied to the object?


vector addition example 2_step 1l


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