Find two unit vectors orthogonal to both given vectors. <3, −3, 3> , <0, 6, 6>

Answer 1

You can first perform the crossproduct to find a vector that is orthogonal to both vectors given. This should give you the vector: <-36, -18, 18>. We need to scale this vector to make it unit length. We do this by dividing by its magnitude.

To find the magnitude: we take √((-36)²+(-18)²+(18²))=18√6.

Thus, out unit length vector parallel to <-36, -18, 18> is <-36/(18√6), -18/(18√6), 18/(18√6)>

=<-√6/3, -√6/6, √6/6> when simplified. To find the other unit vector parallel to this one, just scale by -1

Answer 2

vector <3,-3,3> has magnitude = sqr(9+9+9)= sqr27 = 3sqr3=about 5.2

vector <0,6,6> has magnitude = sqr(36+36) = 6sqr2= about 8.5

3sqr3 x 6sqr2 x sin90 = 18sqr6= about 44.1

perpendicular vector = <-36, -18, 18>= -36i -18j +18k

use matrix algebra

divide by sqr6 to get a unit vector

<-36/sqr6,-18/sqr6,18/sqr6>

rationalize the denominator

=<-6sqr6, -3sqr6, 3sqr6>

is one orthogonal unit vector

multiply by -1 for a 2nd

=<6sqr6, 3sqr3, -3sqr3>

Answer 3

The cross product of these two vectors will be orthogonal to both of them and any vector parallel to that cross product vector will also be orthogonal to both of them.