1 Vectors v1 and v2 are perpendicular and their magnitudes a

1. Vectors v1 and v2 are perpendicular and their magnitudes are known: v1 = 4, v2 = 2. Compute the dot product of vectors w1and w2 if vector w1=4v1 + 5v2 and and w2 = -5v1 +5v2.


2. The length of u is 2, the length of v is 4, and the angle between the two vectors is 60 degrees. Find the length of u +v.

Solution

1.

w1.w2=(4v1+5v2).(-5v1+5v2)=-20||v1||^2+25||v2||^2=-20*4^2+25*2^2=-320+100=-220

2.

||u+v||^2=(u+v).(u+v)=||u||^2+||v||^2+2u.v=2^2+4^2+2||u|| ||v||cos(60 degrees)

                                                                                  =4+16+2*2*4*0.5=20+8=28

Hence,

||u+v||=sqrt{28}

This is the length of u+v