<p>The equation of a line of greatest slope on an inclined plane can be represented as:</p>
<p>(a) $$\frac{x}{3} = \frac{y}{1} = \frac{z}{-1}$$</p>
<p>(b) $$\frac{x}{3} = \frac{y}{-1} = \frac{z}{1}$$</p>
<p>(c) $$\frac{x}{-3} = \frac{y}{1} = \frac{z}{1}$$</p>
<p>(d) $$\frac{x}{1} = \frac{y}{3} = \frac{z}{-1}$$</p>
Step-by-Step Solution
Key Concept: The line of greatest slope is perpendicular to the line of intersection of the two planes. Use cross products to find the direction vector.
Solution: The line of greatest slope lies in plane P _1 and is perpendicular to the line of intersection of P _1 and the horizontal plane P _2. Plane P _1: 4 x - 3 y + 7 z = 0, with normal n _1 = (4, -3, 7) Plane P _2: 2 x + y - 5 z = 0, with normal n _2 = (2, 1, -5) Direction of line of intersection: $\mathbf{n_1} \times \mathbf{n_2} = (4, 17, 5)$ Direction of line of greatest slope: $\mathbf{n_2} \times (\mathbf{n_1} \times \mathbf{n_2}) = (2, 1, -5) \times (4, 17, 5) = (3, -1, 1)$ Since the origin lies on both planes, the equation is: $\frac{x}{3} = \frac{y}{-1} = \frac{z}{1}$
Correct Answer: B