The Dot Product - Scalar Product

Dot product angle

The dot product of two vectors is defined as follows:

A.B = |A| |B| cos(θ), where |A|, |B| = magnitude of A and B,and θ = angle between the two vectors;

In order to see the impact of the change in angle on the dot product of the two vectors below, please move the slider slowly and see the value.
|A| =4; |B| = 6

 

Angle: 00 1800

 

Since the right hand side of the formula is a number, a scalar, this is also called the scalar product.

It is important to note that the angle between the two vectors must be considered in the following way:

Dot product angle

 

 

 

 

 

The Dot Product of Unit Vectors

Dot product unit vectors

i.j = |i| |j| cos(90)
= 1 X 1 X 0
= 0

i.i = |i| |i| cos(0)
= 1 X 1 X 1
= 1

j.j = |j| |j| cos(0)
= 1 X 1 X 1
= 1

k.k = |k| |k| cos(0)
= 1 X 1 X 1
= 1

i.k = |i| |k| cos(90)
= 1 X 1 X 0
= 0

j.k = |j| |k| cos(90)
= 1 X 1 X 0
= 0

If two vectors are parallel, the dot product is maximum.

If two vectors are perpendicular, the dot product is zero.



 

 

E.g.1

If a = 2i + 3j -5k and b = 3i - 5j + 4k, find the a.b.

a.b = (2i + 3j -5k).(3i - 5j + 4k)
= 6i.i - 15j.j -20k.k (i.j = j.k = i.k = 0)
= 6 - 15 - 20
= -29.

 

 

E.g.2

If a = 3i + 2j -5k and b = 4i - 5j + 3k, find the angle between the two vectors.

a.b = (3i + 2j -5k).(4i - 5j + 3k)
= 12i.i - 10j.j -15k.k (i.j = j.k = i.k = 0)
= 12 - 10 - 15
= -23
|a| = √(9 + 4 + 25) = √38; |b| = √(16 + 25 + 9) = √50;
-13 = √38 √50 cos(θ)
cos(θ) = -0.2982
θ = 107.34

 

 

E.g.3

Using the dot product, verify that Pythagoras Theorem is true.

Dot product pythagoras theorem

AC.AC= (AB + BC).(AB + BC)
AC.AC= AB.AB + AB.BC + BC.AB + BC.BC
z z cos(0) = x x.cos(0) + x y cos(90) + x y cos(90) + y y cos(0)
z2 = x2 + 0 + 0 + y2
z2 = x2 + y2
This is Pythagoras Theorem.

 

 

E.g.4

Using the dot product, verify the cosine rule.

Dot product cosine rule

AC.AC= (AB + BC).(AB + BC)
AC.AC= AB.AB + AB.BC + BC.AB + BC.BC
z z cos(θ) = x x.cos(θ) + x y cos(180-θ) + x y cos(180-θ) + y y cos(θ)
z2 = x2 - xy cos(θ) - xy cos(θ) + y2
z2 = x2 + y2 - 2xy cos(θ)
This is the cosine rule.

 

 

E.g.5

Using the dot product, prove that the angle of a semi-circle is a right angle.

Dot product circle theorem

AB.BC= (AO + OB).(BO + OC)
= AO.BO + AO.OC + OB.BO + OB.OC
= r r.cos(180-θ) + r r cos(0) + r r cos(180) + r r cos(θ)
= -r2 cos(θ) + r2 - r2 + r2 cos(θ)
= 0
Since the dot product of AB and BC is zero, AB is perpendicular to BC.

Challenge:

Using the dot product, show that the diagonals of a square intersect at right angles.

 

 

 

 

Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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