Question

Find the direction cosines of the line passing through the points
P (2, 3, 5) and Q (–1, 2, 4).

• A. (-2,-1,-1)
• B. $\left(\frac{3}{\sqrt{11}},\frac{1}{\sqrt{11}},\frac{1}{\sqrt{11}}\right)$
• C. $\left(\frac{-3}{\sqrt{11}},\frac{-1}{\sqrt{11}},\frac{-1}{\sqrt{11}}\right)$
• D. $\left(\frac{-3}{\sqrt{11}},\frac{1}{\sqrt{11}},\frac{1}{\sqrt{11}}\right)$

Answer 1

Answer: (B), (C)

The correct options are
B $\left(\frac{3}{\sqrt{11}},\frac{1}{\sqrt{11}},\frac{1}{\sqrt{11}}\right)$
C $\left(\frac{-3}{\sqrt{11}},\frac{-1}{\sqrt{11}},\frac{-1}{\sqrt{11}}\right)$

The direction cosines of a line passing through the points P $\left(x_1,y_1,z_1\right)$ and
Q $\left(x_2,y_2,z_2\right)$ are
$\frac{x_2-x_1}{\mathrm{PQ}},\frac{y_2-y_1}{\mathrm{PQ}},\frac{z_2-z_1}{\mathrm{PQ}}or\frac{x_1-x_2}{\mathrm{PQ}},\frac{y_1-y_2}{\mathrm{PQ}},\frac{z_1-z_2}{\mathrm{PQ}}\mathrm{Here}\mathrm{PQ}=\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2+(z_1-z_2{)}^2}=\sqrt{(-1-2{)}^2+(2-3{)}^2+(4-5{)}^2}=\sqrt{9+1+1}=\sqrt{11}$
Hence D.C.'s are
$\left(\frac{-3}{\sqrt{11}},\frac{-1}{\sqrt{11}},\frac{-1}{\sqrt{11}}\right)or\left(\frac{3}{\sqrt{11}},\frac{1}{\sqrt{11}},\frac{1}{\sqrt{11}}\right)$