Question

The equation of the line through the point \( (0, 1, 2) \) and perpendicular to the line \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{-2} \] is:

• A. \( \frac{x}{-3} = \frac{y - 1}{4} = \frac{z - 2}{-4} \)
• B. \( \frac{x}{-3} = \frac{y - 1}{4} = \frac{z - 2}{3} \)
• C. \( \frac{x}{-4} = \frac{y - 1}{-4} = \frac{z - 2}{-3} \)
• D. \( \frac{x}{3} = \frac{y - 1}{4} = \frac{z - 2}{-3} \)

Hint:

In geometry, when two lines are perpendicular, their direction ratios must satisfy the condition that their dot product equals zero.

Answer 1

The Correct Option is A

We are given that the line passes through the point \( (0, 1, 2) \) and is perpendicular to the given line. The direction ratios of the given line are \( 2, 3, -2 \), as taken from the coefficients in the equation. Since the lines are perpendicular, the dot product of their direction ratios must be zero. So, we take the direction ratios of the line passing through \( (0, 1, 2) \) as \( a, b, c \). The equation of the line will be: \[ a \cdot 2 + b \cdot 3 + c \cdot (-2) = 0 \] Now, substitute \( a = 2, b = -3, c = 4 \) to check for perpendicularity: \[ (2)(-3) + (3)(4) + (-2)(3) = -6 + 12 - 6 = 0 \] So, we get that the equation of the line is: \[ \frac{x}{-3} = \frac{y - 1}{4} = \frac{z - 2}{-4} \] Thus, the correct equation of the line is option (1).