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: Conceptual Basis:
Collinearity of three points A, B, and C signifies their presence on a single straight line. This geometrical arrangement necessitates that the vector from A to B ($\vec{AB}$) must be parallel to the vector from B to C ($\vec{BC}$). Parallel vectors are characterized by one being a scalar multiple of the other.

Step 2: Methodological Framework:
1. Determine the vectors $\vec{AB}$ and $\vec{BC}$. This is achieved by subtracting the position vector of the initial point from the position vector of the terminal point for each segment.
$\vec{AB} = \vec{OB} - \vec{OA}$
$\vec{BC} = \vec{OC} - \vec{OB}$
2. For 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 ratios of their corresponding vector components must be equal: $\frac{p_x}{q_x} = \frac{p_y}{q_y}$.

Step 3: Execution of Method:
Given position vectors: $\vec{OA} = 20\hat{i} + \lambda\hat{j}$, $\vec{OB} = 5\hat{i} - \hat{j}$, and $\vec{OC} = 10\hat{i} - 13\hat{j}$.
Calculate $\vec{AB}$:
\[ \vec{AB} = \vec{OB} - \vec{OA} = (5\hat{i} - \hat{j}) - (20\hat{i} + \lambda\hat{j}) = (5-20)\hat{i} + (-1-\lambda)\hat{j} = -15\hat{i} - (1+\lambda)\hat{j} \]
Calculate $\vec{BC}$:
\[ \vec{BC} = \vec{OC} - \vec{OB} = (10\hat{i} - 13\hat{j}) - (5\hat{i} - \hat{j}) = (10-5)\hat{i} + (-13 - (-1))\hat{j} = 5\hat{i} - 12\hat{j} \]
As A, B, and C are collinear, $\vec{AB} \parallel \vec{BC}$. 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:
\[ -3 \times 12 = 1 + \lambda \]
\[ -36 = 1 + \lambda \]
Solve for $\lambda$:
\[ \lambda = -36 - 1 = -37 \]

Step 4: Conclusion:
The determined value for $\lambda$ is -37.