18 2k are orthogonal perpendicular Find a third Show that 2i

18. 2]--k are orthogonal (perpendicular). Find a third Show that 2i-j + 4k and 5 vector perpendicular to both.

Solution

solution -:

Two vectors u = (a,b) and v = (c,d) in a coordinate plane are perpendicular if and only if their scalar product a*c + b*d is equal to zero: a*c + b*d = 0.

so we have u=(2,-1,4), and v= ( 5,2,-2)

u *v = ( 2*5 +(-1)*2+4(-2)

=10-2-8

=10-10

=0     so it is orthogonal

To find third vector we will find matrix

[ i j k ] [ 2 -1 4] [ 5 2 -2]

we get by cross product   ( -10, -24,9) as answer

21) B|A| +A|B|= BA+AB= 2AB

A|B|- B|A| = AB- BA = 0

So product of 2AB and 0 is 0 hence orthogonal