Question

The lines \(\overline{\mathrm{r}}=\overline{\mathrm{a}}+\lambda(\overline{\mathrm{b}} \times \overline{\mathrm{c}})\) and \(\overline{\mathrm{r}}=\overline{\mathrm{c}}+\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})\) will intersect if

• A. \(\overset{-}{\mathrm{a}}\times \overset{-}{\mathrm{b}}=\overset{-}{\mathrm{b}}\times \overset{-}{\mathrm{c}}\)
• B. \(\overset{-}{\mathrm{a}}\cdot \overset{-}{\mathrm{b}}=\overset{-}{\mathrm{b}}\cdot \overset{-}{\mathrm{c}}\)
• C. \(\overline{a}\cdot \overline{c}=|\overline{b}|^{2}\)
• D. \(\overline{a}\times \overline{b}=\overline{c}\times \overline{a}\)

Hint:

Two lines intersect when the vector joining the points is perpendicular to the cross product of the directions.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Lines \(\bar{r}=\bar{a}+\lambda\bar{d}_{1}\) and \(\bar{r}=\bar{c}+\mu\bar{d}_{2}\) with \(\bar{d}_{1}=\bar{b}\times\bar{c}\) and \(\bar{d}_{2}=\bar{a}\times\bar{b}\) meet if \((\bar{c}-\bar{a})\cdot(\bar{d}_{1}\times\bar{d}_{2})=0\).

Step 2: Cross Product of Directions
\((\bar{b}\times\bar{c})\times(\bar{a}\times\bar{b})=-[\bar{a}\ \bar{b}\ \bar{c}]\,\bar{b}\).

Step 3: Apply the Condition
\((\bar{c}-\bar{a})\cdot\bar{b}=0\) gives \(\bar{c}\cdot\bar{b}=\bar{a}\cdot\bar{b}\).

Step 4: Result
So the lines intersect if \(\bar{a}\cdot\bar{b}=\bar{b}\cdot\bar{c}\), option (B).

Final Answer:
The lines intersect when \(\bar{a}\cdot\bar{b}=\bar{b}\cdot\bar{c}\). \[ \boxed{\text{(B) }\bar{a}\cdot\bar{b}=\bar{b}\cdot\bar{c}} \]