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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.

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mathsVector AlgebraMultiplication of Vectors by Scalar

Repeated addition of a Vector

In this page repeated addition of vectors is explained.

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We have learned about addition of two or more vectors. Consider a vector being repeatedly added to itself. vector repeated addition What is the result of the addition `vec p + vec p`?

  • sum `= 2 times vec p`
  • sum `= 2 times vec p`
  • component form is required to find the sum
  • directional cosines are required to find the sum
  • sum `=2 times |vec p|`

The answer is 'sum `= 2 times vec p`'. If `vec p = ai+bj+ck` then `vec p + vec p = 2ai+2bj+2ck`. This equals `2 times vec p`.

Repeated addition can be generalized to multiplication of vector by a scalar. Scalar multiplier is denoted with `lambda` to identify differently to the component values `a, b, c`. Note that the scalar multiplier and component values are real numbers. multiplication of vector by scalar What is the result of `lambda vec p` if `vec p = ai+bj+ck`?

  • `lambda ai+bj+ck`
  • `lambda i+ lambda j + lambda k`
  • `lambda ai+ lambda bj + lambda ck`
  • `lambda ai+ lambda bj + lambda ck`
  • `lambda sqrt(a^2+b^2+c^2)`

The answer is '`lambda a i+ lambda b j + lambda c k`'.

When vector is multiplied by a scalar, the vector scales up or down proportionally.

Multiplication of Vector by a scalar: For any vector `vec p = ai + bj+ ck in bbb V` and scalar `lambda in RR`
`lambda vec p= lambda ai+ lambda bj + lambda ck`

Can a vector `vec p` be divided by a real number `lambda`?

  • Yes. Division is inverse of multiplication.
  • Yes. Division is inverse of multiplication.
  • No. Division of vector is not defined.

The answer is 'Yes. Division is inverse of multiplication.'

If the given vector is `vec p = ai+bj+ck` then
`vec p -: lambda`
`quad quad = 1/lambda xx vec p`
`quad quad = a/lambda i+b/lambda j+c/lambda k

slide-show version coming soon