VECTORS:
WHAT ARE THEY,
HOW DO YOU ADD (SUBTRACT) THEM,
AND WHY SHOULD WE CARE?
Dr. Richard A. Thomas
First, it might help to know what a scalar is. A scalar is just a number. Examples of scalars are your age, the number of dollar bills in your wallet, the power (in Watts) of a light bulb, the current speed of the wind, the voltage of a battery, the energy you expend lifting up a book, etc., etc. You add and subtract scalars just like you would numbers. If you have 6 one-dollar bills and your mother gives you 3 more, you would then have 6 + 3 = 9 of them.
Vectors, on the other hand, are composed of two things: 1) a number or length or (more properly) a magnitude, and 2) a direction. Examples of vectors are the current velocity of the wind, the acceleration of your car down highway 494 last night at 9:00 pm, the gravitational force the planet Jupiter exerts on you at this moment, the displacement of a bird as it flies from one tree to another, etc., etc.
Adding vectors is not just like adding scalars. If you have, say, a 24 m displacement vector pointing due north and a 32 m displacement vector pointing due E, the sum of these two vectors is not 24 m + 32 m = 56 m. You must take into account not just the length or magnitude of these vectors, but also their direction. Thus:
The vector you get when you add a 24 m vector N and a 32 m vector E is a 40 m vector that points 36.87o N of E. The way you do this is to put the tail (end without the arrowhead) of the second vector right on the head (end with the arrowhead) of the first vector, and draw a line from the tail of the first to the head of the second:
The angle is obtained using some trigonometry and the length is obtained using the Pythagorean Theorem.
The reason we need to learn how to add vectors is because of the definition of displacement. If we travel N 24 meters and E 32 meters, what is our net displacement? Displacement is a distance “as the crow flies,” so to speak, or a straight-line distance from one point to another. To find this straight-line distance, we need to add displacement vectors. Also, problems involving relative velocity involve adding velocity vectors. Σ F = ma also involves adding vectors. Each force on an object is represented by a vector. The sum of these vectors gives you the acceleration vector times the mass of the object.
Sometimes we need to add two vectors that are not at right angles to each other. Let’s say, for example, we travel SE 40 m and then WNW (22.5o N of W) 20 m. What is our net displacement?
We can certainly draw it out and find the resultant displacement vector:
But what is the exact length of the resultant? What is its exact angle with the horizontal?
To answer these questions we need to split the 40 m and 20 m vectors into components. In other words, we need to define a coordinate system (x and y axes) and find the x and y components of these vectors. Typically one chooses a coordinate system like this
but this is not the only choice. Often it is advantageous to choose one that is tilted with respect to this one. For this problem, we’ll just use the coordinate system above. Choosing and indicating a coordinate system is very important! If you write on a test, “The direction is in the positive x direction,” but nowhere is a coordinate system specified, the answer is wrong.
The x and y components are calculated as follows.
40 m SE:
20 m WNW:
The components were figured out using some trigonometry and/or knowledge of right triangles.
The x-component of the resultant is just the sum of the x-components of the two vectors. Similarly, the y-component of the resultant is just the sum of the y-components of the two vectors.
x-component of resultant = 28.3 m + (-18.5 m) = 9.8 m
y-component of resultant = -28.3 m + 7.7 m = -20.6 m
Using trig and the Pythagorean Theorem, we find that the resultant vector is 23 m long and points 65o S of E. (Here I was feeling compulsive so I stayed with 2 significant figures.) To sum up:
How do I subtract vectors?
There are two methods, and they work equally well. To illustrate them I will subtract the two vectors in the previous example (instead of adding them).
Method A: This is the method I prefer.
First, note that this is the same as
or
(Negative a vector is just the same as that vector pointing in the opposite direction.)
Now this becomes a vector addition problem. As in vector addition, split each vector into x and y components, etc.
x-component of resultant = 28.3 m + 18.5 m = 36.8 m
y-component of resultant = -28.3 m + (- 7.7 m) = -36.0 m
length of resultant = = 51 m (2 significant figures)
angle = arctan (36.0/36.8) = 43.45o.
Method B: An alternative method. You may use any correct method that works for you.
First, put the two vectors tail to tail.
The resultant is a vector that starts at the head of the second and ends at the head of the first:
You will still need to break both vectors into components (like in method A) to figure out the exact length and angle of the resultant.