Prove using vectors that the line joining one vertex of a parallelogram to the midpoint of an opposite side trisects a diagonal of the parallelogram.

Let u=22₁ - 2k -2Ĵ + and = 22-2Ĵ-k. Find a vector w such that the parallelepiped spanned by u, ū and w is a cube. what is the volume of this cube? What is the volume of parallel piped spanned by the vectors ū, ū+v, and ů + V + W ?

Given two non-zero vectors a and b such that abs(a+b)=abs(a-b), find the value of a*b (dot product of a and b).   Abs( is absolute value of. Thank you!

Consider the three vectors = (0, -2,3), v = (1, -4,0) and w = (-4,3,-5). a) Find the cross product vX w. b) Find the area of the parallelogram generated by the vectors and w. Round your answer to 2 decimal places. cl