Question

The projection of a line segment joining the points $P(x_1,y_1,z_1,)$ and $Q(x_1,y_1,z_1,)$ on another line whose DC's are $l,m,n$ is given by

• A. $l(x_1+x_2)+m(y_2+y_2)+n(z_1+z_2)$
• B. $2[\frac{(l{x}_2+m{y}_2+n{z}_2)}{2}-\frac{(l{x}_1+m{y}_1+n{z}_1)}{2}]$
• C. $l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1)$
• D. $\frac{x_2-x_1}{l}+\frac{y_2-y_1}{m}+\frac{z_2-z_1}{n}$

Answer 1

Answer: (B), (C)

The correct options are
B $2[\frac{(l{x}_2+m{y}_2+n{z}_2)}{2}-\frac{(l{x}_1+m{y}_1+n{z}_1)}{2}]$
C $l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1)$

Let the line be of length $P$

D.R.'s of line are $\frac{x_2-x_1}{P},\frac{y_2-y_1}{P},\frac{z_2-z_1}{P}$

D.C.'s of line on which project is taken is $(l,m,n)$

So, projection $=P$[(D.R. of line).(D.R. of line on which project is taken)]

$P[\frac{x_2-x_1}{P}\cdot l+\frac{y_2-y_1}{P}\cdot m+\frac{z_2-z_1}{P}\cdot n]$

$=(x_2-x_1)l+(y_2-y_1)m+(z_2-z_1)n$ ie option C

$=x_{2}l+y_{2}m+z_{2}n-x_{1}l-y_{1}m-z_{1}n$

$=x_{2}l+y_{2}m+z_{2}n-(x_{1}l+y_{1}m+z_{1}n)$

$=2\left(\frac{x_{2}l+y_{2}m+z_{2}n-(x_{1}l+y_{1}m+z_{1}n)}{2}\right)$

$=2(\frac{x_{2}l+y_{2}m+z_{2}n}{2}-\frac{x_{1}l+y_{1}m+z_{1}n}{2})$

ie Option B