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Adding and Subtracting Vectors

Adding and Subtracting Vectors

Vectors can be added and subtracted.

Adding two vectors

Let `vec(AB) = bb(a) = ((4),(4))`, `vec(BC) = bb(b) = ((5),(0))`, and `vec(AC) = bb(c) = ((9),(4))`

Start at A and move along vector a to point B. Then move to point C using vector b. This is equivalent to moving directly from A to C using vector c.

`bb(c) = bb(a) + bb(b) = ((4 + 5),(4 + 0)) = ((9),(4))`

This is the Triangle Law for Vector Addition. This shows that vectors can be added, and from that they can also be subtracted, multiplied and divided.

The Parallelogram Law for Vector Addition states that the diagonal vector, g, is equal to f + e. The diagonal vector is called the resultant vector. You can get the same resultant vector by adding the vectors in the other order. This shows that adding vectors is commutative: it does not matter which way round the vectors are added:

`bb(a) + bb(b) = bb(b) + bb(a)`

Parallelogram Law for Vector Addition.

Example 1

Write a + b as a column vector:

Addition of two vectors

a = `((4),(4))` and b = `((4),(2))`

`((4),(4)) + ((4),(2)) = ((8),(6))`

Answer: `((8),(6))`

Example 2

g = c - d. Write g as a column vector.

Subtraction of two vectors

g = `((3),(3)) - ((-4),(-4)) = ((7),(7))`

Answer: `((7),(7))`