Find a nonzero vector that is perpendicular to both a and b

Find a nonzero vector that is perpendicular to both a and b. a = 2i + 9j - 4k, b = i + j - k

Solution

if there are two vectors a,b then a × b is perpendicular to both a and b.

First compute a × b = determinate of ((i,j,k),(2,9,-4),(1,1,-1))

= (-9+4)i - (-2+4)j + (2-9) k

= -5i -2j -7k

|a × b| = ((-5)²+(-2)²+(-7)²)^0.5

=( 25+4+49)^0.5

   ⁼ 78

the unit vectors perpendicular to given vectors a,b = c = ( -5/ 78 , -2/ 78 , -7 / 78) and ( 5/ 78 , 2/ 78 , 7/ 78 ) .

we can verify this, dot product of perpendicular vectors is 0.

a.c = (-5*2)+(-2*9)+(-7*-4) = 0

b.c = (-5*1)+(-2*1)+(-7*-1)= 0