<p><strong>82.</strong> The equation of parallel line passing through the point \((2, 3, -4)\) and parallel to the vector \(6\hat{i} + 3\hat{j} - 4\hat{k}\) is</p>
Step-by-Step Solution
Key Concept: A line parallel to a given vector and passing through a point has the parametric form: r = r₀ + tᾱ, where r₀ is the position vector of the point and ᾱ is the direction vector. This can also be written in symmetric form as (x-x₀)/a = (y-y₀)/b = (z-z₀)/c.
Step 1: Identify the given data: <ul><li>Point through which line passes: P(2, 3, -4)</li><li>Direction vector: d = 6 î + 3 ĵ - 4 k̂ </li><li>Direction ratios: a = 6, b = 3, c = -4</li></ul> Step 2: Write the parametric form of the line: r = (2, 3, -4) + t(6, 3, -4) Or: x = 2 + 6t, y = 3 + 3t, z = -4 - 4t Step 3: Write the symmetric form (Cartesian equation): $\frac{x-2}{6} = \frac{y-3}{3} = \frac{z-(-4)}{-4}$ Step 4: Simplify: $\frac{x-2}{6} = \frac{y-3}{3} = \frac{z+4}{-4}$ This is the standard equation of the required parallel line.
Correct Answer: 7