Projection Vector???? ):

Question

Q1a) Find the projection p of a = (2,-2,1) in the direction b = (3,-2,-8)

b) Find the projection vector p of a in the direction b

 

Q2a) Find the projection of a = (-2,1,5) onto the line x = (2,2,-1) + t(1,2,-2)

There was a note that said: 'do not take the absolute value of the projection; that is, if the projection is calculated as negative then enter that negative value' ?????

b) Find the projection vector of a onto the line given in part a)

Answer 1

Projection of a = (-2,1,5) onto the line x = (2,2,-1) + t(1,2,-2)

 The points P, (2, 2, 1), and (-2, 1, 5) form a right triangle,along the hypotenuse one has the vector connecting (2, 2, 1) to

(-2, 1, 5); that’s (-2−2, 1-2, 5-1)=(-4,-1,4):

 

 

Projection vector of (-4,-1,4) in the direction (1,2,-2) 

 

Projection of (-4,-1,4) in the direction (1,2,-2) 

 

Answer 2

1)b)

Calculate the dot product:

Calculate the magnitude of a.

|b|=√(3)^2+(-2)^2+(-8)^2 =√9+4+64=√77

 

Calculate the projection vector of a in the direction of b.

proj b (a) = b (a dot b)/(|b|^2)

           =(3,-2,-8)(2)/(√77)^2

             =2/77)(3,-2,-8)

 

1)a) proj b (a) =(a dot b)/|b|

                   = 2/√77