3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
Let for $\lambda \in [0, \infty)$ such that $(x, y, z) \neq (0, 0, 0)$ and $(\vec{i} + \vec{j} + 3\vec{k})x + (3\vec{i} - \vec{j} + \vec{k})y + (4\vec{i} + 5\vec{j})z = \lambda(x\vec{i} + y\vec{j} + 3\vec{k})$, then the value of $\frac{x - y - z}{x}$ is equal to __________.
Step-by-Step Solution
Key Concept: Use the determinant condition to find when lines are coplanar, then solve the system of plane equations to find the line of intersection.
The two lines have direction vectors that give coefficients from the vector equation. Setting up the determinant condition with components and solving yields $\lambda = -1$. The three planes formed by the direction conditions are: $2x + 3y = 4z$, $x - 2y = -5z$, and $3x + y = -z$. Solving the first two equations simultaneously gives $y = 2z$ and $x = -z$, so the line of intersection can be written parametrically with direction ratios determined from these relations.
Correct Answer: I need to find the value of $\frac{x - y - z}{x}$ from the given vector equation.
The equation is:
$$(\vec{i} + \vec{j} + 3\vec{k})x + (3\vec{i} - \vec{j} + \vec{k})y + (4\vec