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

Answer 1

Lily Adams

The answer is #=〈0,1/sqrt2,-1/sqrt2〉#

The cross product yields the vector perpendicular to two other vectors.

#〈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〉#

confirming through the use of 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 vector is divided by the modulus to yield the unit vector.

#=1/(4sqrt2)〈0,4,-4〉#

#=〈0,1/sqrt2,-1/sqrt2〉#

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Answer 2

Charles Aeschliman

The unit vector orthogonal to the plane containing <0, 4, 4> and <1, 1, 1> is <-2/3, 2/3, -1/3>.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

What is the unit vector that is orthogonal to the plane containing # &lt;0, 4, 4&gt; # and # &lt;1, 1, 1&gt; #?

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