What is the unit vector that is orthogonal to the plane containing # <0, 4, 4> # and # <1, 1, 1> #?

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Answer 1 · 2016-12-12T17:00:34

The answer is #=〈0,1/sqrt2,-1/sqrt2〉#

The vector that is perpendicular to 2 other vectors is given by the cross product.

#〈0,4,4〉#x#〈1,1,1〉= | (hati,hatj,hatk), (0,4,4), (1,1,1) | #

#=hati(0)-hatj(-4)+hatk(-4)#

#=〈0,4,-4〉#

Verification by doing the dot products

#〈0,4,4〉.〈0,4,-4〉=0+16-16=0#

#〈1,1,1〉.〈0,4,-4〉=0+4-4=0#

The modulus of #〈0,4,-4〉# is #=∥〈0,4,-4〉∥#

#=sqrt(0+16+16)=sqrt32=4sqrt2#

The unit vector is obtained by dividing the vector by the modulus

#=1/(4sqrt2)〈0,4,-4〉#

#=〈0,1/sqrt2,-1/sqrt2〉#