Practice Adding & Subtracting Vectors

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  • 0:00 Vectors
  • 0:43 Vector Resolution
  • 6:52 Lesson Summary
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Lesson Transcript
Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

Vectors are entities that have two pieces of information associated with them, magnitude and direction. In this lesson, we will practice adding and subtracting vectors both graphically and algebraically. To do this we will break vectors down into their components.


Let's say I need you to draw a one-inch arrow on a piece of paper, and the arrow has to be completely vertical. You've got two pieces of information regarding this arrow. The first piece of information is the length of the arrow, or its magnitude. The second piece of information is the orientation of the arrow, which is how it's oriented in space. An arrow is how you draw the symbol for a vector. A vector is an entity in physics that has a magnitude and a direction. An example of a vector is force, which is how much effort is put into pushing or pulling on something (its magnitude), and in which direction the force acts.

Vector Resolution

Many times in physics we have to add vectors both numerically and graphically. We subtract them, too, which is really just adding a negative. Let's go through a few examples showing how to add and subtract vectors using both methods. But, before we can do that, we have to be able to resolve a vector. Resolving a vector means to calculate the x, y, and z components of the vector. To do this, we use the trigonometric functions: sine, cosine, and tangent. The formula for these functions is given in the table on your screen below. As you can see:

Table 1

As you can see in the equation on your screen below, we use i-hat, j-hat, and k-hat to represent the unit vectors in the x-direction, y-direction, and z-direction, respectively.

Unit vectors

Vectors are the hypotenuses of right triangles, unless they are in a pure direction like along the x, y, or z axes. As you can see in the diagram on your screen below, a vector is shown in the x-y plane, along with its component vectors.

Diagram 1

The angle θ can be determined using the tangent function we just discussed a moment ago.


The reason we have to resolve vectors is because we can only add x-vectors to x-vectors, y-vectors to y-vectors, and z-vectors to z-vectors. It is similar to algebra, where you can only add like terms. For example, in the equation 2x + 3y + 4x = 5, we can only add the 2x and the 4x giving us 6x + 3y = 5.

Adding vectors graphically involves three rules.

Rule 1: Do not change the magnitudes or orientation of any vectors.

Rule 2: Put the tip of one vector (the arrow part) to the tail of the other vector.

Rule 3: Draw the resultant arrow from the tail of the first vector to the tip of the last vector.

Let's go through some examples involving vector addition.

Here's the prompt for our first example: Add vectors R1 and R2, both numerically and graphically.


Here's our solution: Graphically adding these vectors we get what you can see on your screen below.


The tail of the red vector was put on the tip of the purple vector. The black vector is the resultant vector.

Now we'll use the numerical process for adding these vectors. Since the vectors are drawn on a grid, we can determine the x and y components of each vector without using trigonometry.


Adding the vectors numerically, we get:


Now we can graphically show that both results are same.


Notice that we put the tail of the y-component of vector 1 to the tip of vector 2, which is only a y-component. It is in the pure y-direction.

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