#### notes

Suppose a vector `vec (AB)` makes an angle θ with a given directed line l (say), in the anticlockwise direction in following fig.

Then the projection of `vec (AB)` on l is a vector `vec p` (say) with magnitude `|vec (AB)| |cos θ| ` , and the direction of `vec p` being the same (or opposite) to that of the line l, depending upon whether cosθ is positive or negative. The vector `vec p`

is called the projection vector, and its magnitude `|vec p|` is simply called as the projection of the vector `|vec (AB)|` on the directed line l.

For example , in each of the following fig. , projection vectors of `vec (AB)` along the line l is vector `vec (AC).`

**Observations: **

1) If `hat p` is the unit vector along a line l , then the projection of a vector `vec a` on the line l is given by `vec a . hat p.`

2) Projection of a vector `vec a` on other vector `vec b`, is given by

`vec a . hat b,` or `vec a ((vec b)/(|vec b|))` , or `1/|vec b| (vec a.vec b)`

3) If θ = 0, then the projection vector of `vec(AB)` will be `vec (AB)` itself and if θ = π, then the projection vectors of `vec (AB)` will be `vec (BA)`.

4) If `θ = π / 2 or θ = (3π)/2`, then the projection vector of `vec (AB)` will be zero vector.

**Remark:** If α, β and γ are the direction angles of vector `vec a = a_1 hat i + a_2hat j + a_3 hat k` , then its direction cosines may be given as

`cos alpha = (vec a . hat i)/(|vec a||hat i|) = a_1/|vec a|` , `cos β = a_2/ |vec a|` and `cos gamma = a_3 /|vec a|`

Also , note that `|vec a| cos alpha,` `|vec a| cos β ` and `|vec a| cos gamma` are respectively the projections of `vec a` along OX, OY and OZ i.e. the scalar components `a_1,a_2` and `a_3` of the vector `vec a`, are precisely the projections of `vec a` along x - axis , y- axis and z - axis , respectively. Further , if `vec a` is a unit vector , then it may be expressed in terms of its direction cosines as

`vec a = cos alpha hat i + cos β hat j + cos gamma hat k `

Video link : https://youtu.be/qmZmuB553hY