Let u and v be vectors of length 4 and 3 respectively and su

Let u and v be vectors of length 4 and 3, respectively, and suppose that Find (u - v) (2u - 3v) and ||u + u||. The two vectors u + v and u - 3v are orthogonal. Find the angle between u and u if ||u|| =2||v||. Given the points A(-1,0) and B(2,3) in the plane, find all points C such that A, B and C are the vertices of a right triangle with a right angle at A and with Prove that (ac + 3bd)^2

Solution

4) given length of u=4 , v=3 and u.v = 8

(u -v) . (2u -3v) = 2u^2 - 3v.u -2v.u + 3v^2

= 2(u)^2 -5u.v +3v^2

= 2(4)^2 -5*8 +3(3)^2

= 2(16) - 40 +3(9)

=32 -40 +27

=59 -40

=19

to find|u+v|

square |u+v|

|u+v|^2 = u^2 +v^2 +2u.v

= 4^2 +3^2 +2 *8

=16 +9+16

|u+v|^2 =41

then |u+v| = sqrt(41)

b) . given u +v and u -3v are othgonal

then (u+v) . (u-3v) =0

u.u -3v.u + v.u -3v^2

and given |u|= 2|v|

4v^2 -2u.v -3v^2 =0

-2(2v) .v cos(theta) = -v^2

-4v^2 cos(theta) = -v^2

cos(theta) =1/4

theta = arc cos(1/4)