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

$=1/(4sqrt2)〈0,4,-4〉$

$=〈0,1/sqrt2,-1/sqrt2〉$

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