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mathsVector AlgebraProperties of Cross Product

When products of two vectors are equal

In this page, you will learn the property when two vector cross products are equal

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

  • `x=a`
  • `x=b`
  • `x=b`
  • `x=ab`

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 an unknown `vec x`. Given that `vec x xx vec p = vec q xx vec p`. What is the value of `vec x`?vector cross product no cancellation Note: All the vectors shown in yellow-dotted-line will form parallelograms of same area with `vec p`.

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

The answer is 'Cannot be calculated'.

`vec x xx vec p = vec q xx 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 cross product no cancellation Note that in cross product, `vec q` is split into orthogonal components and the component in perpendicular to `vec p` is in the product. The component parallel to `vec p` is lost.

 •  Given cross products are equal, does not imply the vectors are equal.

Cannot Cancel: Given `vec x xx vec p = vec q xx 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 xx vec p = vec q xx vec p` imply that the component perpendicular to `vec p` of both `vec x` and `vec q` are equal.vector cross product no cancellation If we subtract `vec x - vec q` then the common component will cancel out and the remaining vector will be parallel to `vec p`.

`vec x xx vec p = vec q xx vec p`

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

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

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

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

slide-show version coming soon