NIMCET 2022 — Mathematics PYQ

We use the Vector Triple Product formula:

$\vec{x}\times (\vec{y}\times \vec{z})=(\vec{x}\cdot \vec{z})\vec{y}-(\vec{x}\cdot \vec{y})\vec{z}$

Step 1: Expand the Left Hand Side (LHS)

For $(\vec{a}\times \vec{b})\times \vec{c}$, we treat it as $-\vec{c}\times (\vec{a}\times \vec{b})$:

Step 2: Expand the Right Hand Side (RHS)

Step 3: Equate LHS and RHS

Step 4: Simplify the equation

The common term $(\vec{a}\cdot \vec{c})\vec{b}$ cancels out from both sides:

Step 5: Interpretation

The equation $(\vec{b}\cdot \vec{c})\vec{a}=(\vec{a}\cdot \vec{b})\vec{c}$ shows that vector $\vec{a}$ is a scalar multiple of vector $\vec{c}$ (since the dot products are just scalar numbers).

When one vector is a scalar multiple of another ($\vec{a}=k\vec{c}$), the vectors are collinear.

Final Answer:

The vectors $\vec{a}$ and $\vec{c}$ are collinear. (Option C)