Question: The angle which the line $\frac{x}{1} = \frac{y}{-1} = \frac{z}{0}$ makes with the positive direction of Y-axis is:
Step 1 (Formula / Governing Law & Given Data): [0.25 Mark]
The direction ratios of the given line are $(a, b, c) = (1, -1, 0)$. To find the angle $\theta$ with the Y-axis, we first determine the direction cosines $(l, m, n)$ of the line. The angle $\theta$ with the Y-axis is given by the relation:
$$\cos \theta = \frac{|b|}{\sqrt{a^2 + b^2 + c^2}}$$However, to find the specific angle with the positive direction of the Y-axis, we use the direction cosine $m$:
$$\cos \theta = m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}$$Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]
- Calculate the magnitude of the direction vector: $$\sqrt{a^2 + b^2 + c^2} = \sqrt{(1)^2 + (-1)^2 + (0)^2} = \sqrt{1 + 1 + 0} = \sqrt{2}$$
- Calculate the direction cosine $m$ corresponding to the Y-axis: $$m = \frac{b}{\sqrt{a^2 + b^2 + c^2}} = \frac{-1}{\sqrt{2}}$$
- Set up the equation for the angle $\theta$: $$\cos \theta = -\frac{1}{\sqrt{2}}$$
- Since $\cos \theta$ is negative, $\theta$ lies in the second quadrant: $$\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$$
Step 3 (Final Concluded Answer): [0.25 Mark]
The angle which the line makes with the positive direction of the Y-axis is $\frac{3\pi}{4}$. Therefore, the correct option is (B).
💡 EL10 Concept Insight: Think of the direction ratios as a set of coordinates for a vector; because the Y-component is negative ($-1$), the line points "downward" relative to the Y-axis, which forces the angle to be obtuse (greater than 90 degrees) rather than acute.