Refresher - Basic Definitions

Basic Definitions

Additions of Vectors

$(a_{1}, a_{2}) + (b_{1}, b_{2}) = (a_{1} + b_{1}) (a_{2} + b_{2})$

Scalar Multiple of a Vector

$m (a_{1}, a_{2}) = (m a_{1}, m a_{2})$

Properties of Addition and Scalar Multiplication of Vectors

Bold letters denote vectors.

For any vectors a, b, c and scalars m, n.

Where 0 = < 0, 0>

If the vector A and the angle $θ$ are defined as above, then

$A_{x} = | | A | | cos θ and A_{y} = | | A | | sin θ$ .

The Dot Product

Let a $= (a_{1}, a_{2}) = a_{1} i + a_{2}$ and b $= (b_{1}, b_{2}) = b i + b_{2} j$ . The dot product of a and b, denoted by $a • b$ , is

$a • b = (a_{1}, a_{2}) • (b, b_{2}) = a_{1} b_{1} + a_{2} b_{2}$ .

The symbol $a • b$ is read “a dot b”.

Definition of Parallel and Orthogonal Vectors

Let $θ$ Be the angle between two nonzero vectors u and v.

u and v are parallel if $θ = 0$ ,or $θ = π$ .
u and v are orthogonal if $θ = \frac{π}{2}$ . Alternatively, two vectors u and v are orthogonal if and only if u .v = 0

If $θ$ is the angle between two nonzero vectors u and v, then
u .v $= | | u | | | | v | | cos θ$ .