NIMCET 2011 — Mathematics PYQ

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Question

NIMCET 2011 — Mathematics PYQ

NIMCET | Mathematics | 2011

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are unit coplanar vectors, then the scalar triple product $[2\mathbf{a} - \mathbf{b}, 2\mathbf{b} - \mathbf{c}, 2\mathbf{c} - \mathbf{a}]$ will be equal to:

Choose the correct answer:

Explanation

1. Condition for Coplanar Vectors:

Since $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are coplanar vectors, their scalar triple product is zero:

[\mathbf{a}, \mathbf{b}, \mathbf{c}] = 0

2. Property of Scalar Triple Product:

For any vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$ defined as linear combinations of $\mathbf{a}, \mathbf{b}, \mathbf{c}$ :

[l_1\mathbf{a} + m_1\mathbf{b} + n_1\mathbf{c}, \,\, l_2\mathbf{a} + m_2\mathbf{b} + n_2\mathbf{c}, \,\, l_3\mathbf{a} + m_3\mathbf{b} + n_3\mathbf{c}] = \begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix} [\mathbf{a}, \mathbf{b}, \mathbf{c}]

3. Applying the property to the given expression:

Let $\mathbf{V} = [2\mathbf{a} - \mathbf{b}, 2\mathbf{b} - \mathbf{c}, 2\mathbf{c} - \mathbf{a}]$

\mathbf{V} = \begin{vmatrix} 2 & -1 & 0 \\ 0 & 2 & -1 \\ -1 & 0 & 2 \end{vmatrix} [\mathbf{a}, \mathbf{b}, \mathbf{c}]

4. Calculation:

Substituting $[\mathbf{a}, \mathbf{b}, \mathbf{c}] = 0$ :

\mathbf{V} = \begin{vmatrix} 2 & -1 & 0 \\ 0 & 2 & -1 \\ -1 & 0 & 2 \end{vmatrix} \times 0

\mathbf{V} = 0

Correct Option: (a) $0$