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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.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

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`

                            
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