Project Using the Transformation

Problem

u=[[1],[−3],[4]],v=[[2],[−3],[1]],w=[[−1],[4],[3]]

Solution

1. Identify the vectors in standard component form. The vector u is given as a column vector, while v and w are given in terms of unit vectors i j and k

2. Calculate the dot product v⋅w to be used in the projection formula.

v⋅w=(2)*(−1)+(−3)*(4)+(1)*(3)

v⋅w=−2−12+3=−11

3. Calculate the magnitude squared of the vector onto which we are projecting, |w|2

|w|2=(−1)2+(4)2+(3)2

|w|2=1+16+9=26

4. Apply the formula for the vector projection of v onto w denoted as (proj_w)*v

(proj_w)*v=(v⋅w)/(|w|2)*w

(proj_w)*v=(−11)/26*[[−1],[4],[3]]

5. Distribute the scalar to find the final components of the projected vector.

(proj_w)*v=[[11/26],[−44/26],[−33/26]]

Final Answer

(proj_w)*v=[[11/26],[−22/13],[−33/26]]

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