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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

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In this page, you will learn about the bilinear property of vector cross product.

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Cross product is defined between `vec p, vec q in bbb V` and the result `vec p xx vec q in bbb V`. This can be considered as a transformation. What type of transformation is this?

  • unary
  • binary
  • binary
  • ternary

The answer is 'binary' as the input is two vectors. 'bi' means two.

Which of the following property is true for Bilinear transformation?

  • `T(x+y, z)=T(x,z)+T(y,z)`
  • `T(ax,z)=aT(x,z)`
  • `T(ax+y, z)= aT(x,z)+T(y,z)`
  • All the above
  • All the above

The answer is 'All the above'.

The property `T(ax+y, z)= aT(x,z)+T(y,z)` defines linearity for binary operator `T`. That is why it is named Bi-linear.

Consider the cross product as a binary transform `T(vec p, vec q) = vec p xx vec q`. Is cross product bilinear?

  • Yes
  • Yes
  • No

The answer is 'Yes'. The cross product is distributive over vector addition and also satisfies the scalar multiple property.

 •  Cross product is bilinear.

Bilinear Property: For any vector `vec p, vec q, vec r in bbb V` and `lambda in RR`
`(lambda vec p + vec q) xx vec r = lambda (vec p xx vec r) + (vec q xx vec r)`

Solved Exercise Problem:

Given `vec x xx vec z = 2` and `vec y xx vec z = 1`, what is `(3 vec x + 2 vec y) xx vec z`?

  • `5`
  • `6`
  • `8`
  • `8`
  • `9`

The answer is '`8`'
`(3 vec x + 2 vec y) xx vec z`
`quad = 3 vec x xx vec z + 2 vec y xx vec z`

slide-show version coming soon