Question

The equation of a line passing through $(3, -1, 2)$ and perpendicular to the lines $\bar{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda (2\hat{i} - 2\hat{j} + \hat{k})$ and $\bar{r} = (2\hat{i} + \hat{j} - 3\hat{k}) + \mu (\hat{i} - 2\hat{j} + 2\hat{k})$ is

• A. $\frac{x - 3}{2} = \frac{y + 1}{3} = \frac{z - 2}{2}$
• B. $\frac{x - 3}{3} = \frac{y + 1}{2} = \frac{z - 2}{2}$
• C. $\frac{x + 3}{2} = \frac{y + 1}{3} = \frac{z - 2}{2}$
• D. $\frac{x - 3}{2} = \frac{y + 1}{2} = \frac{z - 2}{3}$

Hint:

Check the point coordinates first! A line passing through $(3, -1, 2)$ must have a numerator sequence of $(x-3)$, $(y+1)$, and $(z-2)$. This instantly eliminates option (C).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to determine the Cartesian equation of a line in 3D space.
The line passes through a designated coordinate point $(x_1, y_1, z_1) = (3, -1, 2)$ and is orthogonal to two existing lines.
The direction vectors of the given lines can be extracted directly from their vector equations as the coefficients of the scalar parameters $\lambda$ and $\mu$.

Step 2: Key Formula or Approach:
The symmetric Cartesian equation of a line passing through $(x_1, y_1, z_1)$ with direction ratios $(a, b, c)$ is:
$$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$ Since the required line is perpendicular to two lines with direction vectors $\bar{b}_1$ and $\bar{b}_2$, its direction vector $\bar{b}$ must be parallel to their cross product:
$$\bar{b} = \bar{b}_1 \times \bar{b}_2$$

Step 3: Detailed Explanation:
Extract the direction vectors of the two given lines:
$$\bar{b}_1 = 2\hat{i} - 2\hat{j} + \hat{k} \implies (2, -2, 1)$$ $$\bar{b}_2 = \hat{i} - 2\hat{j} + 2\hat{k} \implies (1, -2, 2)$$ Compute the cross product using a determinant to find the direction ratios $(a, b, c)$:
$$\bar{b}_1 \times \bar{b}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -2 & 1 \\ 1 & -2 & 2 \end{vmatrix}$$ Expand the determinant along the first row:
$$\bar{b}_1 \times \bar{b}_2 = \hat{i}[(-2)(2) - (1)(-2)] - \hat{j}[(2)(2) - (1)(1)] + \hat{k}[(2)(-2) - (-2)(1)]$$ $$\bar{b}_1 \times \bar{b}_2 = \hat{i}[-4 + 2] - \hat{j}[4 - 1] + \hat{k}[-4 + 2]$$ $$\bar{b}_1 \times \bar{b}_2 = -2\hat{i} - 3\hat{j} - 2\hat{k}$$ The direction ratios are $(-2, -3, -2)$, which can be scaled by multiplying by $-1$ to give $(2, 3, 2)$.
Substitute the coordinates of the given point $(3, -1, 2)$ and the direction ratios into the standard formula:
$$\frac{x - 3}{2} = \frac{y - (-1)}{3} = \frac{z - 2}{2} \implies \frac{x - 3}{2} = \frac{y + 1}{3} = \frac{z - 2}{2}$$

Step 4: Final Answer:
The Cartesian equation of the line is $\frac{x - 3}{2} = \frac{y + 1}{3} = \frac{z - 2}{2}$, corresponding to option (A).