The line $\mathrm{L}_{1}$ is parallel to the vector $\overrightarrow{\mathrm{a}}=-3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and passes through the point $(7,6,2)$ and the line $\mathrm{L}_{2}$ is parallel to the vector $\vec{b}=2 \hat{i}+\hat{j}+3 \hat{k}$ and passes through the point $(5,3,4)$. The shortest distance between the lines $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ is:

JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Three Dimensional Geometry

The line

L_{1}

is parallel to the vector

a = - 3 \hat{i} + 2 \hat{j} + 4 \hat{k}

and passes through the point

(7, 6, 2)

and the line

L_{2}

is parallel to the vector

b = 2 \hat{i} + \hat{j} + 3 \hat{k}

and passes through the point

(5, 3, 4)

. The shortest distance between the lines

L_{1}

and

L_{2}

is:

Answer: (A)

With

b_{1} = (- 3, 2, 4)

and

b_{2} = (2, 1, 3)

b_{1} \times b_{2} = (2, 17, - 7)

and

∣ b_{1} \times b_{2} ∣ = 4 + 289 + 49 = 342 = 338

. The joining vector is

a_{2} - a_{1} = (5 - 7, 3 - 6, 4 - 2) = (- 2, - 3, 2)

, and

(a_{2} - a_{1}) \cdot (b_{1} \times b_{2}) = - 4 - 51 - 14 = - 69

. The shortest distance is

\frac{∣ - 69∣}{3 38} = \frac{23}{38}

JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Single correct option