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Thought-Process to Discover Knowledge

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

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mathsVector AlgebraProperties of Dot Product

Dot product of a scalar multiple

In this page, you will learn about the vector dot product with the scalar multiple of a vector.



click on the content to continue..

Given `vec p = 2i+j-k` and `lambda = 3.1` what is the scalar multiple `3.1 vec p`?

  • `6.2i+3.1j-3.1k`
  • `6.2i+3.1j-3.1k`
  • `6.2i+j-k`

The answer is '`6.2i+3.1j-3.1k`'

Given the definition of dot product as
`vec p cdot vec q = |vec p||vec q|cos theta`
Dot product of a scalar multiple What is `vec p cdot (lambda vec q)`?

  • `lambda(vec p cdot vec q)`
  • `|p||lambda q|cos(theta)` if `lambda` is positive
  • `lambda|p||q|cos(theta)`
  • all the above
  • all the above

The answer is 'All the above'

Note that if `lambda` is a negative number, it equals `|p||lambda q|cos(180+theta)`.

Given the definition of dot product as
`vec p cdot vec q = p_xq_x+p_yq_y+p_zq_z`
What is `vec p cdot (lambda vec q)`?

  • `lambda (vec p cdot vec q)`
  • `p_x(lambda q_x)+p_y(lambda q_y)+p_z(lambda q_z)`
  • `lambda (p_xq_x+p_yq_y+p_zq_z)`
  • all the above
  • all the above

The answer is 'All the above'

 •  Dot product with a scalar multiple of a vector equals the scalar multiple of the dot product with the vector.

Dot Product of scalar multiple: For any vectors `vec p, vec q`
`vec p cdot (lambda vec q) = lambda (vec p cdot vec q)`

Solved Exercise Problem:

Given that `vec p cdot vec q = 3j+1.1k`, what is `vec p/(2.1) cdot vec q`?

  • `3/(2.1)j+1.1/(2.1)k`
  • `3/(2.1)j+1.1/(2.1)k`
  • `6.3j+2.31k`
  • `3/(2.1)j+1.1k`

The answer is '`3/(2.1)j+1.1/(2.1)k`'.

                            
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