What is the unit vector that is orthogonal to the plane containing (8i+12j+14k) and (2i+j+2k)?

1 Answer
David G.
Aug 6, 2016

Two steps are required:

  1. Take the cross product of the two vectors.
  2. Normalise that resultant vector to make it a unit vector (length of 1).

The unit vector, then, is given by:

(10√500i+12√500j−16√500k)

Explanation:

  1. The cross product is given by:

(8i+12j+14k)×(2i+j+2k)
=((12⋅2−14⋅1)i+(14⋅2−8⋅2)j+(8⋅1−12⋅2)k)
=(10i+12j−16k)

  1. To normalise a vector, find its length and divide each coefficient by that length.

r=√102+122+(−16)2=√500≈22.4

The unit vector, then, is given by:

(10√500i+12√500j−16√500k)

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