# Current Affairs JEE Main & Advanced

## Projection

Category : JEE Main & Advanced

Projection of a line joining the points$\mathbf{P(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{y}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{z}}_{\mathbf{1}}}\mathbf{)}$ and $\mathbf{Q(}{{\mathbf{x}}_{\mathbf{2}}}\mathbf{,}{{\mathbf{y}}_{\mathbf{2}}}\mathbf{,}{{\mathbf{z}}_{\mathbf{2}}}\mathbf{)}$ on another line whose direction cosines are $\mathbf{l,}\,\,\mathbf{m}$ and $\mathbf{n}$ : Let PQ be a line segment where $P\equiv ({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})$ and $Q\equiv ({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})$ and AB be a given line with d.c.’s as $l,\,\,m,\,\,n$. If the line segment PQ makes angle $\theta$  with the line AB, then

Projection of PQ is $P'Q'=PQ\cos \,\theta$

$=({{x}_{2}}-{{x}_{1}})\cos \alpha +({{y}_{2}}-{{y}_{1}})\cos \beta +({{z}_{2}}-{{z}_{1}})\cos \gamma$

$=({{x}_{2}}-{{x}_{1}})l+({{y}_{2}}-{{y}_{1}})m+({{z}_{2}}-{{z}_{1}})n$.

For x-axis,$l=1,\,\,m=0,\,\,n=0$.

Hence, projection of PQ on x-axis $={{x}_{2}}-{{x}_{1}}$.

Similarly, projection of PQ on y-axis $={{y}_{2}}-{{y}_{1}}$ and projection of PQ on  z-axis $={{z}_{2}}{{z}_{1}}$.

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