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mathsVector AlgebraProperties of Dot Product

### When products of two vectors are equal

In this page, you will learn whether vectors can be canceled in both side of the equation when two vector dot products are equal.

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Consider real numbers a, b in RR and an unknown number x. Given that ax = ab, what is the value of x?

• x=a
• x=b
• x=b
• x = a b

The answer is 'x=b'. Both the left hand side and right hand side of the equation is divided by a to arrive at the solution.

Consider vectors vec p, vec q in bbb V and a unknown vec x. Given that vec x cdot vec p = vec q cdot vec p. What is the value of vec x? Note: All the vectors shown in yellow-dotted-line will have the same projection on to the vec p.

• cannot be calculated
• cannot be calculated
• vec x = vec q

The answer is 'Cannot be calculated'.

vec x cdot vec p = vec q cdot vec p does not imply that vec x = vec q.
That is, vec p cannot be canceled on left-hand-side and right-hand-side. Note that in dot product, vec q is split into orthogonal components and the component in parallel to vec p is only in the product. The component perpendicular to vec p is lost.

•  Equal dot products does not imply the vectors are equal.

Cannot Cancel: Given vec x cdot vec p = vec q cdot vec p does not imply vec x = vec q. That is, the vec p cannot be canceled on both sides of the equation or on numerator and denominator in a division.

vec x cdot vec p = vec q cdot vec p imply that both vec x and vec q has same projection on to vec p shown as a If we subtract vec x - vec q then the common component will cancel out and the remaining vector will be perpendicular to vec p.

vec x cdot vec p = vec q cdot vec p

Subtracting vec q cdot vec p from both the sides. vec x cdot vec p - vec q cdot vec p = vec q cdot vec p - vec q cdot vec p
(vec x - vec q)cdot vec p = (vec q - vec q)cdot vec p
(vec x - vec q)cdot vec p = 0 cdot vec p
(vec x - vec q)cdot vec p = 0

The above can be understood as vector vec x - vec q is orthogonal to vec p.

•  If dot products are equal then difference of the vectors will be perpendicular to the vector with which dot products are equal.

Subtraction on sides of an Equation: vec x cdot vec p = vec q cdot vec p imply that
(vec x - vec q)cdot vec p = 0
Which implies vec x - vec q is either vec 0 or is perpendicular to vec p.

slide-show version coming soon