The direction cosines of the line passing through the points (1, 2, −3) and (3, −1, 3) are
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{2}{7}, -\frac{3}{7}, \frac{6}{7}\)
The direction ratios of the line through \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are \(x_2-x_1,\ y_2-y_1,\ z_2-z_1\):
\(3-1 = 2,\quad -1-2 = -3,\quad 3-(-3) = 6\)
Dividing by \(\sqrt{a^2+b^2+c^2}\) gives the direction cosines:
\(\sqrt{4+9+36} = \sqrt{49} = 7\)
\( \displaystyle l = \frac{2}{7},\quad m = -\frac{3}{7},\quad n = \frac{6}{7} \)
The opposite direction gives \(-\frac{2}{7}, \frac{3}{7}, -\frac{6}{7}\), which is not among the options. Options (1), (2) and (3) each change the sign of only some of the three values, so they are not direction cosines of this line.