Vector algebra so far has allowed us to add and subtract vectors, to multiply a vector by a scalar and to find the length of a vector. What about some sort of multiplication? Let's learn about it in A-Level Maths!

There are, in fact, two types of multiplication with vectors. Here we will discuss one of them.

**Defination**

**a · b = |a| × |b| × cos(θ) **

where a and b are the magnitudes/length of a and b, and theta is the angle between the two vectors.

You see that the result of this multiplication is a scalar (not a vector), so it’s known as the scalar product.

Because the symbol for this multiplication is a dot, it’s also known as the dot product.

**Why cos(θ) ?**

To multiply two vectors it makes sense to multiply their lengths together but only when they point in the same direction. So we make one "point in the same direction" as the other by multiplying by cos(θ):

It works exactly the same if we "projected" b along side a then multiplied:

Because it doesn't matter which order we do the multiplication:

**|a| × ****|b| × cos(θ)**** = ****|a| × cos(θ)**** × |b|**

**Properties **

a**·**b = b**·**a - the dot product is commutative

[This is easily seen from the definition because the elements in ab cos θ are ordinary numbers whose multiplication is commutative.]

a·a = a^{2}

a·b = 0 <---> if a is perpendicular to b (cos 90° = 0 )

The big one:

= axi·i …

=ax + by + cz

If you dot one of the unit vectors with itself, the answer is 1 x 1 x cos 0 = 1.

If you dot two unlike unit vectors, the answer is 1 x 1 x cos 90° = 0

**Examples:**

What is the acute angle between the vectors

4 + 10 + 18 = 32

But it’s also

So θ = 12.9°

N.B.

When two lines cross, they form four angles: two equal acute angles and two equal obtuse angles.

If the calculation gives a negative sign for cosθ, then that will give you the obtuse angle. If you want the acute angle, you just subtract your answer from 180°.

### Calculate the dot product of vectors a and b:

a · b = |a| × |b| × cos(θ)

a · b = 10 × 13 × cos(59.5°)

a · b = 10 × 13 × 0.5075...

a · b = 65.98... = 66 (rounded)

OR we can calculate it this way:

a · b = a_{x} × b_{x} + a_{y} × b_{y}

a · b = -6 × 5 + 8 × 12

a · b = -30 + 96

a · b = 66

Drafted by Eunice (Maths)

Reference

https://www.mathsisfun.com/algebra/vectors-dot-product.html