How do you find the scalar, vector and orthogonal projections?

Let a = (-4, -4, -10) and b = (-4, 9, 6) be vectors. Find the scalar, vector, and orthogonal projections of b onto a.|||The scalar projection is just b鈭檃/||a||. The vector projection is the product of this scalar with a unit vector in the direction of a. So the vector projection is





proj_a(b) = (b鈭檃/||a||虏) a





and the orthogonal projection is the difference b - proj_a(b)----this allows you to write b as the sum of two vectors, one parallel to a (the projection onto a) and one perpendicular.





a鈭檅 = 16 - 36 - 60 = -80





||a|| = 2鈭?33).





So the scalar projection is -80/(2鈭?33)) = -40/鈭?33). The vector projection is








proj_a(b) = -80/(2鈭?33))虏 (-4, -4, -10) = (-20/33)(-4, -4, -10)





and





b - proj_a(b) = 1/33 (-212, 217, -2).