JEE Main · 2005 · 08 May 2005 · Offline · Vector Algebra

Let

a, b

and

c

be distinct non-negative numbers. If the vectors

a \hat{i} + a \hat{j} + c \hat{k}, \hat{i} + \hat{k}

and

c \hat{i} + c \hat{j} + b \hat{k}

lie in a plane, then

c

Answer: (A)

Solution

The three vectors are coplanar when their scalar triple product vanishes:

a 1 c a 0 c c 1 b = 0

. Expanding along the first row, this is

a (0 \cdot b - 1 \cdot c) - a (1 \cdot b - 1 \cdot c) + c (1 \cdot c - 0 \cdot c) = - a c - a (b - c) + c^{2} = c^{2} - ab

. Setting it to zero gives

c^{2} = ab

, so

c

is the geometric mean of

a

and

b

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option