What is Basic Vector Math? | Definition, Types, Formula & Examples

What is Basic Vector Math?

A vector is an attribute that has both magnitude and direction. Vectors are drawn as an arrow with a tail and a head. The length of the vector represents its magnitude.

Vectors are written in letters and bold. For example, you will have vector a or vector b. If you were just talking about the magnitude of the vector, you would write the letter inside the parallel lines like this: ||a||

Adding Vectors

Vectors can be added to find the result of two vectors (a + b = c). Direction and magnitude are combined when adding vectors. Here are some simple examples that add vectors in the same direction or 180 degrees from the same (negative) direction.

What do we do when we are adding vectors that are not in the same direction?

Head-to-Tail Method

One way to add vectors is to use the head-to-tail method. In this method, we place the tail of the additional vector at the end of the beginning of the previous vector. The resulting vector is the vector drawn from the tail of the first vector to the beginning of the last vector.

Pythagorean Theorem

If two vectors a and b form a 90 degree angle, we can use the Pythagorean theorem to find the magnitude of the resulting vector c. You can go here to learn more about the Pythagorean theorem.

In this case, the magnitude of the sum vector a + b = c is a2 + b2 = c2.

Example problem:

Jim went four miles north, then three miles east. What is the distance covered if he travels a straight line from the beginning to the end?

Since Jim follows two vectors, one to the north and one to the east, we can add these vectors to get the answer. Since north and east are 90 degrees apart, we can use the Pythagorean theorem.

c2 = a2 + b2
c2 = 32 + 42
c2 = 9 + 16
c2 = 25
c = 5

Commutative Law

The law of commutativity for vector addition states that the order in which the vectors are added does not matter.

a + b = b + c

Associative Law

The law of association for vector addition states that when three or more vectors are added, it doesn’t matter which vector is added first.

(a + b) + d = a + (b + d)

Subtracting Vectors

When you subtract two vectors a – b, it is the same as adding vectors a + (-b). Negative vectors have the same magnitude but opposite direction as positive vectors.