Let a=(0,2,3) and b=(-5,5,-1). Find the projection of b onto a.

Answer 1

projection b onto a =vector p=[(bdota)/|a|] times a/|a|

a/|a|=(0/√13,2/√13,3/√13)

(bdota)/|a|=7/√13

p=(0,14/13,21/13)

it is kind of neat

yes bdota=b_xa_x+b_ya_y+b_za_z=value=scalar constant

in this case 7

but also bdota=(|b|)(|a|)cosθ ,,,,,where θ is angle between the vectors b and a

(it is astonishing a bit to see since it is cosine,,, theta angle being positive or negative is not relevant,

so bdota=adotb)

so one can see in geometry dividing bdota by |a| which is |b|cosθ

is projecting a hypotenuse value |b| onto line formed by a

To make projection a vector onto a must multiply value times unit vector a/|a|

doing many of these it is easier to remember

p=[(bdota)/|a|²]times the vector a

this way no square root is needed for |a|

Answer 2

Projection of b onto a is b·a/¦a¦

¦a¦ is magnitude of a.

a=(0,2,3) and b=(-5,5,-1)

¦a¦=(0² + 2² + 3²)^0.5 or 3.605

b·a=0×-5 + 2×5 + 3×-1 or 7

Projection of b onto a=7/3.605 or 1.94