3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade None
Question:
MATCH THE FOLLOWING:
(A) The plane $x - 2y + 7z + 21 = 0$ contains the line
(B) An equation of the line passing through $3\mathbf{i} - 5\mathbf{j} + 7\mathbf{k}$ and perpendicular to the plane $3x - 4y + 5z = 8$ is ($\lambda, \mu$ are parameters)
(C) Equation of the line of shortest distance between the lines $\frac{x}{2} = \frac{y}{-3} = \frac{z}{1}$ and $\frac{x - 2}{3} = \frac{y - 1}{-5} = \frac{z + 2}{2}$ is
(D) The line of intersection of the planes $\mathbf{r}\cdot(3\mathbf{i} - \mathbf{j} + \mathbf{k}) = 1$ and $\mathbf{r}\cdot(\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}) = 2$ is parallel to the line given by
The plane $x - 2y + 7z + 21 = 0$ contains the line
An equation of the line passing through $3mathbf{i} - 5mathbf{j} + 7mathbf{k}$ and perpendicular to the plane $3x - 4y + 5z = 8$ is ($lambda, mu$ are parameters)
Equation of the line of shortest distance between the lines $ rac{x}{2} = rac{y}{-3} = rac{z}{1}$ and $ rac{x - 2}{3} = rac{y - 1}{-5} = rac{z + 2}{2}$ is
The line of intersection of the planes $mathbf{r}cdot(3mathbf{i} - mathbf{j} + mathbf{k}) = 1$ and $mathbf{r}cdot(mathbf{i} + 4mathbf{j} - 2mathbf{k}) = 2$ is parallel to the line given by
Step-by-Step Solution
Key Concept: A line in 3D space can be characterized by a point it passes through and a direction vector; for plane-line relationships, use point and direction substitution; for line intersections, use cross products of normal/direction vectors.
For (A): A line lies in a plane if both a point on the line and the direction vector satisfy the plane equation. For (B): A line perpendicular to plane $3x - 4y + 5z = 8$ has direction ratios $(3, -4, 5)$, so the parametric form is $\mathbf{r} = 3\mathbf{i} - 5\mathbf{j} + 7\mathbf{k} + \lambda(3\mathbf{i} - 4\mathbf{j} + 5\mathbf{k})$. For (C): The line of shortest distance between two skew lines is perpendicular to both, with direction vector equal to the cross product of their direction vectors. For (D): The line of intersection of two planes is perpendicular to both normal vectors, so its direction is parallel to $\mathbf{n}_1 imes \mathbf{n}_2 = (3\mathbf{i} - \mathbf{j} + \mathbf{k}) imes (\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}) = 2\mathbf{i} + 7\mathbf{j} + 13\mathbf{k}$.
Correct Answer: I need to match each statement (A), (B), (C), (D) with appropriate options. However, the options are not provided in the question. Based on the step-by-step solution given, let me work through each part:
**(A) The plane x - 2y + 7z + 21 = 0 contains the line**