The direction cosines of the line passing through $P\left(2, 3, -1\right)$ and the origin are

Given points are

$p\left(2,3,-1\right)=\left(x_1,y_1,z_1\right)$ and

$\left(0,0,0\right)=\left(x_2,y_2,z_2\right)$

Direction cosines of a line passing through these two points is given by

$\cos \alpha =\frac{x_2-x_1}{\sqrt{{x_1}^2+{y_1}^2+{z_1}^2}}$

$\cos \beta =\frac{y_2-y_1}{\sqrt{{x_1}^2+{y_1}^2+{z_1}^2}}$

$\cos \gamma =\frac{z_2-z_1}{\sqrt{{x_1}^2+{y_1}^2+{z_1}^2}}$

$\cos \alpha =\frac{0-2}{\sqrt{2^2+3^2+{\left(-1\right)}^2}}=\frac{-2}{\sqrt{14}}$

$\cos \beta =\frac{0-3}{\sqrt{2^2+3^2+{\left(-1\right)}^2}}=\frac{-3}{\sqrt{14}}$

$\cos \gamma =\frac{0-(-1)}{\sqrt{2^2+3^2+{\left(-1\right)}^2}}=\frac{1}{\sqrt{14}}$

Hence, the direction cosines are

$\frac{-2}{\sqrt{14}}, \frac{-3}{\sqrt{14}}, \frac{1}{\sqrt{14}}$