Let a, b, c be distinct non-negative numbers. If the vectors ai + aj + ck, i + k and ci + cj + bk lie in a plane, then c is

Let a, b, c be distinct non-negative numbers. If the… — JEE Main 1993 Mathematics PYQ

1993

Let

a, b, 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

A.
the arithmetic mean of $a$ and $b$
B.
the geometric mean of $a$ and $b$
C.
the harmonic mean of $a$ and $b$
D.
equal to zero

Show answer and solution

Correct option: B

Solution

For the three vectors to be coplanar, the scalar triple product must be zero. Setting up the determinant:

|a a c|
|1 0 1| = 0
|c c b|

Expanding: a(0·b - 1·c) - a(1·b - 1·c) + c(1·c - 0·c) = 0

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

= -ac - ab + ac + c² = 0

= c² - ab = 0

So c² = ab, which means c is the geometric mean of a and b.

[1993]

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JEE Main 1993 Mathematics question on Vector Algebra