Question

The equation of the plane passing through the points having position vectors \((\overset{̄}{a}+\overset{̄}{b}),(\overset{̄}{b}+\overset{̄}{c})\) and \((\overset{̄}{a}+\overset{̄}{c})\) is \(\ldots\)

• A. \(\overset{̄}{r}\cdot (\overset{̄}{a}\times \overset{̄}{b}+\overset{̄}{b}\times \overset{̄}{c}+\overset{̄}{c}\times \overset{̄}{a}) = [\overset{̄}{a} \overset{̄}{b} \overset{̄}{c}]\)
• B. \(\overset{̄}{r}\cdot (\overset{̄}{a}\times \overset{̄}{b}+\overset{̄}{b}\times \overset{̄}{c}+\overset{̄}{c}\times \overset{̄}{a}) = 2[\overset{̄}{a} \overset{̄}{b} \overset{̄}{c}]\)
• C. \(\overset{̄}{r}\cdot (\overset{̄}{a}\times \overset{̄}{b}+\overset{̄}{b}\times \overset{̄}{c}+\overset{̄}{a}\times \overset{̄}{c}) = [\overset{̄}{a} \overset{̄}{b} \overset{̄}{c}]\)
• D. \(\overset{̄}{r}\cdot (\overset{̄}{a}\times \overset{̄}{b}+\overset{̄}{b}\times \overset{̄}{c}+\overset{̄}{a}\times \overset{̄}{c}) = 2[\overset{̄}{a} \overset{̄}{b} \overset{̄}{c}]\)

Hint:

Find a normal using two in-plane vectors, then write r . n = (a+b) . n.

Answer 1

The Correct Option is B

Step 1: Verify that all three points satisfy option (B):
Let $\bar n = \bar a\times\bar b + \bar b\times\bar c + \bar c\times\bar a$ and $T = [\bar a\,\bar b\,\bar c]$.
Point $\bar a + \bar b$: $\bar a\cdot\bar n = \bar a\cdot(\bar b\times\bar c) = T$ and $\bar b\cdot\bar n = \bar b\cdot(\bar c\times\bar a) = T$, so total $2T$.
Point $\bar b+\bar c$: $\bar b\cdot\bar n = T$ and $\bar c\cdot\bar n = \bar c\cdot(\bar a\times\bar b) = T$, so total $2T$.
Point $\bar a+\bar c$: $\bar a\cdot\bar n = T$ and $\bar c\cdot\bar n = T$, total $2T$.

Step 2: Conclusion:
All three points give the constant $2T$, so the plane is $\bar r\cdot\bar n = 2T$. Option (A) would need the value $T$, which none of the points gives.

Final Answer:
Option (B). \[ \boxed{\bar r\cdot(\bar a\times\bar b+\bar b\times\bar c+\bar c\times\bar a)=2[\bar a\,\bar b\,\bar c] \text{ (B)}} \]