ˆ ˆ a b k i a b k a b k k (Using the above Properties and ) = a b + a b + a b (Using Observation ) Thus . . Projection of a vector on a line Suppose a vector makes an angle θ with a given directed line l (say), in the anticlockwise direction (Fig . ).
Then the projection of on l is a vector (say) with magnitude , and the direction of being the same (or opposite) to that of the line l , depending upon whether cos θ is positive or negative. The vector Fig . is called the projection vector , and its magnitude | | is simply called as the projection of the vector on the directed line l . For example, in each of the following figures (Fig .
(i) to (iv)), projection vector of along the line l is vector Observations . If ˆ p is the unit vector along a line l , then the projection of a vector on the line l is given by ˆ p . Projection of a vector on other vector r b , is given by ˆ, b or . If θ = , then the projection vector of will be itself and if θ = π , then the projection vector of will be .
If = π θ or = π θ , then the projection vector of will be zero vector. Remark If α , β and γ are the direction angles of vector a i a j a k , then its direction cosines may be given as Also, note that are respectively the projections of along OX, OY and OZ. i.e., the scalar components a , a and a of the vector , are precisely the projections of along x -axis, y -axis and z -axis, respectively. Further, if is a unit vector, then it may be expressed in terms of its direction cosines as cos cos cos α +