Question

Find the angles made by the straight line passing through the points(1,-3,2) and (3,-5,1) with the coordinate axes

Answer 1

The angles made by straight line with coordinate x-axis , y-axis and z-axis   are $( cos^{-1} (\frac{-2}{3} ), cos^{-1} (\frac{-2}{3} ), cos^{-1} (\frac{1}{3} ))$ respectively.

Step-by-step explanation:

Given points are (1,-3,2) and (3,-5,1)

The direction cosine of a line which passes through the points $(x_1,y_1,z_1)$ and  $(x_2,y_2,z_2)$ is   $(\frac{x_1-x_2}{d},\frac{y_1-y_2}{d},\frac{z_1-z_2}{d})$      

 where     $d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2$

Here $d =\sqrt{(1-3)^2+(-3+5)^2+(2-1)^2}$

            $=\sqrt{9}$ =3

Therefore the direction cosine of the line joining by two given points is

$=(\frac{-2}{3}, \frac{-2}{3},\frac{1}{3} )$

Let the straight line makes angles α , β and γ with x-axis , y-axis and z-axis respectively.

Then

$cos\alpha = \frac{-2}{3} \Leftrightarrow \alpha = cos^{-1} (\frac{-2}{3} )$

$cos \beta =\frac{-2}{3} \Leftrightarrow \beta = cos^{-1}(\frac{-2}{3} )$

$cos \gamma =\frac{1}{3} \Leftrightarrow \gamma = cos^{-1}(\frac{1}{3} )$

The angles made by straight line with coordinate x-axis , y-axis and z-axis   are $( cos^{-1} (\frac{-2}{3} ), cos^{-1} (\frac{-2}{3} ), cos^{-1} (\frac{1}{3} ))$ respectively.