Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
Line $L_1$ is parallel to a vector $\vec{a} = 3\hat{i} + 2\hat{j} + 4\hat{k}$ and passes through a point $A(7,6,2)$ and the line $L_2$ is parallel to a vector $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$ and passes through a point $B(5, 3, 4)$. Now a line $L_3$ parallel to a vector $\vec{c} = 2\hat{i} + 2\hat{j} + \hat{k}$ intersects the lines $L_1$ and $L_2$ at points $C$ and $D$ respectively then $|\overrightarrow{CD}|$ is_______.
Step-by-Step Solution
Key Concept: Parallel vectors have proportional components; solve the resulting system of linear equations to find unknown parameters.
Given position vectors of C and D, we find $\vec{CD} = \vec{r_2} - \vec{r_1} = c(2\vec{i} + 2\vec{j} + \vec{k})$. Since $\vec{CD} \parallel \vec{r}$, we equate coefficients from the parallel condition with the given constraint $3a - 2b - 2c + 2 = 0$, $2a - b - 2c + 3 = 0$, and $4a - 3b - c - 2 = 0$. Solving this system yields $a = 2, b = 1, c = 3$, giving $|\vec{CD}| = 3\sqrt{4 + 4 + 1} = 9$.
Correct Answer: 9