NIMCET 2021 — Mathematics PYQ

Step 1: Simplify the first line equation.

The equation is $\vec{r}\times \vec{a}=\vec{b}\times \vec{a}$.

Rearranging terms:

If the cross product of two non-zero vectors is zero, they must be parallel. Therefore:

Substituting the given vectors:

Step 2: Simplify the second line equation.

The equation is $\vec{r}\times \vec{b}=\vec{a}\times \vec{b}$.

Rearranging terms:

Similarly, this implies:

Substituting the given vectors:

Step 3: Find the point of intersection.

At the point of intersection, the position vector $\vec{r}$ must be the same for both lines:

Comparing the coefficients of $\hat{i},\hat{j},\text{ and }\hat{k}$:

1. For $\hat{j}$: $\lambda =1$

2. For $\hat{k}$: $-1=-\mu ⟹\mu =1$

3. For $\hat{i}$: Check consistency $⟹2+\lambda =1+2\mu$

   $$2+1=1+2(1)⟹3=3$$

   (The values are consistent).

Step 4: Calculate the coordinates.

Substitute $\lambda =1$ into the first line equation (or $\mu =1$ into the second):

The coordinates of the point of intersection are $(3,1,-1)$.

Final Answer:

The point of intersection is $(3,1,-1)$.