Question

If the points A, B, C with position vectors $20\hat{i} + \lambda\hat{j}$, $5\hat{i} - \hat{j}$ and $10\hat{i} - 13\hat{j}$ respectively are collinear, then the value of $\lambda$ is

• A. 12
• B. -37
• C. 37
• D. -12

Hint:

For collinearity problems in 2D, setting the ratios of components equal is often the fastest method. Another way is to set the determinant of a matrix formed by the coordinates (with an extra column of 1s) to zero, which represents the area of the triangle formed by the points being zero.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
If three points A, B, and C are collinear, it means they lie on the same straight line. This implies that the vector from A to B ($\vec{AB}$) must be parallel to the vector from B to C ($\vec{BC}$). Two vectors are parallel if one is a scalar multiple of the other.
Step 2: Key Formula or Approach:
1. Find the vectors $\vec{AB}$ and $\vec{BC}$ by subtracting the position vectors of the initial points from the terminal points. $\vec{AB} = \vec{OB} - \vec{OA}$ $\vec{BC} = \vec{OC} - \vec{OB}$ 2. For two vectors $\vec{p} = p_x\hat{i} + p_y\hat{j}$ and $\vec{q} = q_x\hat{i} + q_y\hat{j}$ to be parallel, the ratio of their corresponding components must be equal: $\frac{p_x}{q_x} = \frac{p_y}{q_y}$.
Step 3: Detailed Explanation:
Let the position vectors be $\vec{OA} = 20\hat{i} + \lambda\hat{j}$, $\vec{OB} = 5\hat{i} - \hat{j}$, and $\vec{OC} = 10\hat{i} - 13\hat{j}$.
First, calculate the vector $\vec{AB}$:
\[ \vec{AB} = \vec{OB} - \vec{OA} = (5\hat{i} - \hat{j}) - (20\hat{i} + \lambda\hat{j}) = (5-20)\hat{i} + (-1-\lambda)\hat{j} = -15\hat{i} - (1+\lambda)\hat{j} \] Next, calculate the vector $\vec{BC}$:
\[ \vec{BC} = \vec{OC} - \vec{OB} = (10\hat{i} - 13\hat{j}) - (5\hat{i} - \hat{j}) = (10-5)\hat{i} + (-13 - (-1))\hat{j} = 5\hat{i} - 12\hat{j} \] Since A, B, and C are collinear, $\vec{AB}$ is parallel to $\vec{BC}$. We can equate the ratios of their components:
\[ \frac{\text{x-component of } \vec{AB}}{\text{x-component of } \vec{BC}} = \frac{\text{y-component of } \vec{AB}}{\text{y-component of } \vec{BC}} \] \[ \frac{-15}{5} = \frac{-(1+\lambda)}{-12} \] \[ -3 = \frac{1+\lambda}{12} \] Multiply both sides by 12:
\[ -3 \times 12 = 1 + \lambda \] \[ -36 = 1 + \lambda \] \[ \lambda = -36 - 1 = -37 \] Step 4: Final Answer:
The value of $\lambda$ is -37.