1 Let L be the line in R3 that consists of all scalar multip

1. Let L be the line in R^3 that consists of all scalar multiples of the vector [2 1 2]. Find the orthogonal projection of the vector [1 1 1] onto L.

2. Find the matrices of the linear transformations from R^3 to R^3.

The transformation about the z-axis through an angle of pi/2 counterclockwise as viewed from the positive z- axis.

Solution

Unitize [2 1 2] to get [2/sqrt(5) 1/sqrt(5) 2/sqrt(5)]
Take the dot product of [1 1 1] with [2/sqrt(5) 1/sqrt(5) 2/sqrt(5)]
to get 6/sqrt(5)
=2.683281574
Multiply by [2/sqrt(5) 1/sqrt(5) 2/sqrt(5)]
to get
[12/5 6/5 12/5]