JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Vector Algebra

Let the vectors

(2 + a + b) \hat{i} + (a + 2 b + c) \hat{j} - (b + c) \hat{k}

(1 + b) \hat{i} + 2 bj - b \hat{k}

and

(2 + b) \hat{i} + 2 bj + (1 - b) \hat{k}, a, b, c \in R

be co-planar. Then which of the following is true?

Answer: (B)

Solution

Three vectors are coplanar when their scalar triple product vanishes:

2 + a + b 1 + b 2 + b a + 2 b + c 2 b 2 b - (b + c) - b 1 - b = 0

The three minors of the first row are

2 b (1 - b) + 2 b^{2} = 2 b, (1 + b) (1 - b) + b (2 + b) = 1 + 2 b, 2 b (1 + b) - 2 b (2 + b) = - 2 b

so expansion along the first row gives

(2 + a + b) (2 b) - (a + 2 b + c) (1 + 2 b) + (- (b + c)) (- 2 b) = 0

4 b + 2 ab + 2 b^{2} - a - 2 ab - 2 b - 4 b^{2} - c - 2 b c + 2 b^{2} + 2 b c = 0

Every term carrying

ab

b^{2}

b c

cancels, leaving

2 b - a - c = 0

, that is

2 b = a + c

JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Single correct option