Scalar Product
Vectors can be added and subtracted and also multiplied by a constant. Vectors can also be multiplied.
The scalar product, or dot product, of two vectors a and b, in two or three dimensions, is defined as the product of their lengths and the cosine of the angle between their directions. It is written a.b and in words is said as "a dot b".
a.b = ab cos θ
The angle θ can be acute or obtuse.
Properties of Scalar Products

Scalar products are commutative. That is a.b = b.a

a.a = a^{2}
This is because the angle between vector a and vector a is 0^{o} and cos 0^{o} is equal to 1. 
If neither a nor b is the zero vector then a and b are perpendicular if a.b = 0
This is because the angle between perpendicular vectors is 90^{o} and cos 90^{o} is equal to 0. 
If the value of a.b is known then the angle between the two eactors can be found as:
It can also be shown that if
Examples
Question

Answer

If 
c.d = 2 x 4 + 3 x 8 + 0 x 4 = 8 + 24 + 0 = 32 
The scalar product of c and d is found in two different ways. c.d = 2 x 4 + 3 x 8 + 0 x 4 Therefore, 32 = √13.√96.cosθ The angle between c and d is 25.1^{o} 
The Ratio Theorem
This theorem is very useful for finding the coordinates of a point that divides a line into a certain ratio.
If P is any point on a line AB, and if, a, b and p are the position vectors of A, B and P respectively, then
p = λa + μb
and the ratio AP : PB =μ: λ for the constants λ and μ such that μ + λ =1
μ = AP/(AP + PB)
and
λ =PB/(AP + PB)_
Example
Question

Answer

Point A has coordinates (4, 7) and point B has coordinates (6, 2). Mark M, the midpoint of AB. What is the position vector, OM. 
a. Using p = λa + μb where 
Division of a Line
The table below shows examples of internally and externally dividing a line.
Internal Division

External Division

Point P dividing AB internally in the ratio 1:2 In problems, to find the ratios for λ and μ: μ = 1/(1+2) = 1/3 λ = 2/(1+2) = 2/3 Note that λ + μ = 1

Point P dividing AB externally in the ratio 1:3. (This can also be said as 1:3, the negative sign indicates the division is external) In problems, to find the ratios for λ and μ: μ = 1/(1+1) = 1/2 λ = 3/(1+1) = 3/2 Note that λ + μ = 1 
In general , if P divides AB in the ratio m : l then