What is the unit vector that is normal to the plane containing # ( i +k) # and # ( i + 7 j + 4 k) #?

1 Answer
Eddie
Aug 3, 2016

#hat v = 1/(sqrt(107)) * ((7),(3),(-7))#

Explanation:

first, you need to find the vector (cross) product vector, #vec v#, of those 2 co-planar vectors, as #vec v# will be at right angles to both of these by definition:

#vec a times vec b = abs(vec a) abs(vec b) \ sin theta \ hat n_{color(red)(ab)}#

computationally, that vector is the determinant of this matrix, ie

#vec v = det((hat i, hat j , hat k),(1,0,1),(1,7,4))#

#= hat i (-7) - hat j (3) + hat k (7)#

#= ((-7),(-3),(7))# or as we are only interested in direction
#vec v = ((7),(3),(-7))#

for the unit vector we have

#hat v = (vec v)/(abs (vec v))= 1/(sqrt(7^2 + 3^3 + (-7)^2)) * ((7),(3),(-7))#

#= 1/(sqrt(107)) * ((7),(3),(-7))#

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