Let $\vec{a}=\hat{i}-\hat{k}, \vec{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}$ and $\vec{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}$. Then $[\vec{a}, \vec{b}, \vec{c}]$ depends on

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

Let

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

and

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

. Then

[a, b, c]

depends on

Answer: (D)

The scalar triple product is the determinant

1 x y 01 x - 1 1 - x 1 + x - y

. Expanding along the first row,

1 \cdot [1 \cdot (1 + x - y) - x (1 - x)] - 0 + (- 1) \cdot (x \cdot x - 1 \cdot y) = (1 + x - y - x + x^{2}) - (x^{2} - y) = 1

. The value is

1

whatever

x

and

y

are, so it depends on neither

x

nor

y

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