1.3 Vector Algebra
We can perform a limited amount of algebra on vectors: adding, subtracting, scalar
multiplication, simplfying by collecting and combining like terms, multiplying out
brackets, and solving linear equations. These techniques contribute to the usefulness
of vectors in applications.
Example
collecting
&
combining
like terms
In a vector expression like 2a + b + a + 3b − 4a, the terms 2a, a and −4a
are called like terms because each is a scalar multiple of the vector a. We can
combine like terms by adding their coefficients.
2a + a − 4a = (2 + 1 − 4)a = −a
In a similar way we can combine the multiples of b.
We simplify 2a + b + a + 3b − 4a by collecting and combining like terms, in
exactly the same way as we do in ordinary algebra:
2a + b + a + 3b − 4a = −a + 3b.
Example
expanding
brackets
Vector expressions may also contain brackets such as 2(a + 3b). These can be
multiplied out as in ordinary algebra.
2a + b − 2(a + 3b) = 2a + b − 2a − 6b = −5b
Example
solving
vector
equations
We can solve vector equations in almost the same way that we solve ordinary
linear equations6
, the main difference being that it is customary to multiply
but not divide vectors by scalars.
Solve the equation 2x + a − b + c = 2a − 3b + c for the unknown vector x.
Answer
2x + a − b + c = 2a − 3b + c
2x = a − 2b . . . now multiply both sides by 1
2
.
x =
1
2
(a − 2b) or 1
2
a − b
As you can see, the methods we use for simplifying linear expressions in ordinary algebra carry intuitively over to vectors. However, vectors are different from numbers,
and we should confirm that what we are doing is valid.
6A linear equation is an equation in which the unknown only occurs to the first power.
