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Thought-Process to Discover Knowledge

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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`?vector dot product no cancellation 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. vector dot product no cancellation 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` vector dot product no cancellation 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`.

                            
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