NIMCET 2010 — Mathematics PYQ

1. Find the vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$:

To find the vector between two points, subtract the position vector of the initial point from the position vector of the terminal point.

• For $\overrightarrow{AB}$:

  $$\overrightarrow{AB}=\text{Position Vector of }B-\text{Position Vector of }A$$
  $$\overrightarrow{AB}=(2\hat{i}+5\hat{j}+0\hat{k})-(\hat{i}+\hat{j}+\hat{k})$$
  $$\overrightarrow{AB}=(2-1)\hat{i}+(5-1)\hat{j}+(0-1)\hat{k}$$
  $$\overrightarrow{AB}=\hat{i}+4\hat{j}-\hat{k}$$

• For $\overrightarrow{CD}$:

  $$\overrightarrow{CD}=\text{Position Vector of }D-\text{Position Vector of }C$$
  $$\overrightarrow{CD}=(\hat{i}-6\hat{j}-\hat{k})-(3\hat{i}+2\hat{j}-3\hat{k})$$
  $$\overrightarrow{CD}=(1-3)\hat{i}+(-6-2)\hat{j}+(-1-(-3))\hat{k}$$
  $$\overrightarrow{CD}=-2\hat{i}-8\hat{j}+2\hat{k}$$

2. Analyze the relationship between the two vectors:

Let's look at the components of $\overrightarrow{CD}$:

Factor out $-2$:

Notice that $(\hat{i}+4\hat{j}-\hat{k})$ is exactly the vector $\overrightarrow{AB}$.

So, we have:

3. Determine the angle:

When one vector is a negative scalar multiple of another ($\overrightarrow{V_1}=k\overrightarrow{V_2}$ where k < 0), the two vectors are anti-parallel. This means they point in exactly opposite directions.

• If k > 0, the angle is $0$.

• If k < 0, the angle is $180^{\circ }$ or $\pi$ radians.

Since $k=-2$, the angle between $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is $\pi$.

Correct Option:

(d) $\pi$