Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The given line passes through the points $A(-4,2,-1)$ and $B(-3,1,-2)$.

The direction ratios of the line are proportional to $(-3-(-4),1-2,-2-(-1))$, which simplifies to $(1,-1,-1)$.

The direction cosines of the line are given by $\pm \left(\frac{1}{\sqrt{1^2+(-1{)}^2+(-1{)}^2}},\frac{-1}{\sqrt{3}},\frac{-1}{\sqrt{3}}\right)=\pm \left(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{-1}{\sqrt{3}}\right)$.

It is given that the line makes an acute angle $\gamma$ with the positive direction of the Z-axis, which means $\cos \gamma >0$.

To satisfy this condition, we must choose the direction cosines such that the z-component is positive. Multiplying by $-1$, we get the direction cosines as $\left(\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$.

Thus, the value of $\cos \gamma$ is $\frac{1}{\sqrt{3}}$.

Answer: $\frac{1}{\sqrt{3}}$