If four distinct points with position vectors a⃗, b⃗, c⃗ and d⃗ are coplanar, then [a⃗ b⃗ c⃗] is equal to :

Solution

Maths sir · 3 steps

Coplanar points mean the three edges from a⃗ have zero box product. Expand it and match the signs.

1. Coplanarity condition. The four points are coplanar exactly when the edges b⃗-a⃗, c⃗-a⃗, d⃗-a⃗ lie in one plane. That means their scalar triple product is zero.

   [b⃗-a⃗ c⃗-a⃗ d⃗-a⃗] = [b⃗ c⃗ d⃗] - [a⃗ b⃗ c⃗] + [a⃗ b⃗ d⃗] + [a⃗ d⃗ c⃗] = 0

2. Expand and rearrange. Any term with a⃗ appearing twice is zero, which leaves four terms. Swapping two vectors flips the sign, and a cyclic shift keeps it. So [a⃗ b⃗ c⃗] = [b⃗ c⃗ d⃗] + [a⃗ b⃗ d⃗] + [a⃗ d⃗ c⃗].

   Option	Term 1	Term 2	Term 3
   (a)	[d⃗ b⃗ a⃗] = -[a⃗ b⃗ d⃗]	[a⃗ c⃗ d⃗] = -[a⃗ d⃗ c⃗]	[d⃗ b⃗ c⃗] = [b⃗ c⃗ d⃗]
   (b)	[b⃗ c⃗ d⃗]	[d⃗ a⃗ c⃗] = -[a⃗ d⃗ c⃗]	[d⃗ b⃗ a⃗] = -[a⃗ b⃗ d⃗]
   (c)	[a⃗ d⃗ b⃗] = -[a⃗ b⃗ d⃗]	[d⃗ c⃗ a⃗] = [a⃗ d⃗ c⃗]	[d⃗ b⃗ c⃗] = [b⃗ c⃗ d⃗]
   (d)	[d⃗ c⃗ a⃗] = [a⃗ d⃗ c⃗]	[b⃗ d⃗ a⃗] = [a⃗ b⃗ d⃗]	[c⃗ d⃗ b⃗] = [b⃗ c⃗ d⃗]

3. So. Only (d) gives all three terms with a plus sign: each one is a cyclic shift of a term we need. The answer is (d).

The idea to keep. For coplanar points, set [b⃗-a⃗ c⃗-a⃗ d⃗-a⃗] = 0 and expand. Then check each option term by term: a cyclic shift keeps the sign, and a swap flips it.

Answer from the NTA final answer key. Solution by PrepWiser.