Product of Two Vectors - Projection of a Vector on a Line




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).`

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 `
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