If a = i - k, b = xi + j + (1-x)k and c = yi + xj + (1+x-y)k. Then, [a b c] depends on

If a = i - k, b = xi + j + (1-x)k and c = yi + xj +… — JEE Main 2001 Mathematics PYQ

2001

\vec{a} = \hat{i} - \hat{k}

\vec{b} = x\hat{i} + \hat{j} + (1-x)\hat{k}

and

\vec{c} = y\hat{i} + x\hat{j} + (1+x-y)\hat{k}

. Then,

[\vec{a} \ \vec{b} \ \vec{c}]

depends on

A.
only $y$
B.
only $x$
C.
neither $x$ nor $y$
D.
both $x$ and $y$

Show answer and solution

Correct option: C

Solution

Compute the scalar triple product [a b c] = a · (b × c), equivalently the determinant:

|1, 0, -1|
|x, 1, 1-x|
|y, x, 1+x-y|

Expanding along the first row:
= 1·[1·(1+x-y) - (1-x)·x] - 0 + (-1)·[x·x - 1·y]
= [(1+x-y) - x(1-x)] + (-1)[x² - y]
= [1 + x - y - x + x²] + [-x² + y]
= 1 + x² - y - x² + y
= 1

The scalar triple product equals 1, independent of both x and y.

[2001]

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