Question

If the position vectors of three points $A,B$ and $C$ are respectively i + j + k, 2i + 3j – 4k and 7i + 4j + 9k, then the unit vector to the plane containing the triangle $ABC$ is

• A. 31i – 18j – 9k
• B. $\frac{31i - 38j - 9\ k}{\sqrt{2486}}$
• C. $\frac{31i + 18j + 9k}{\sqrt{2486}}$
• D. None of these

Answer 1

Answer: (B)

$\frac{31i - 38j - 9\ k}{\sqrt{2486}}$

Answer 2

Unit vector perpendicular to plane of $\Delta ABC$ is, $\frac{\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}}{|\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}|}$,

where $\overset{\rightarrow}{AB} = i + 2j - 5k$ and $\overset{\rightarrow}{AC} = 6i + 3j + 8k$

∴ $\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC} = 31i - 38j - 9k$and $|\overset{\rightarrow}{AB} \times \overset{\rightarrow}{AC}| = \sqrt{2486}$

∴ Required vector = $\frac{31i - 38j - 9k}{\sqrt{2486}}$.