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/sqrt500i+12/sqrt500j-16/sqrt500k)#

Explanation:

  1. The cross product is given by:

#(8i+12j+14k) xx (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=sqrt(10^2+12^2+(-16)^2)=sqrt500~~22.4#

The unit vector, then, is given by:

#(10/sqrt500i+12/sqrt500j-16/sqrt500k)#

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