Class CBSE Class 12 Mathematics Vector Algebra Q #1367
COMPETENCY BASED
UNDERSTAND
3 Marks 2025 AISSCE(Board Exam) SA
During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by $\vec{B}=2\hat{i}+8\hat{j}$, $\vec{W}=6\hat{i}+12\hat{j}$ and $\vec{F}=12\hat{i}+18\hat{j}$ respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.
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Detailed Solution

Step 1: Understanding the Section Formula

The section formula states that if a point $P$ divides the line segment joining points $A$ and $B$ with position vectors $\vec{a}$ and $\vec{b}$ respectively in the ratio $m:n$, then the position vector of $P$ is given by: $$ \vec{p} = \frac{m\vec{b} + n\vec{a}}{m+n} $$

Step 2: Applying the Section Formula to the Given Problem

Let the wicketkeeper divide the line segment joining the bowler and the leg slip fielder in the ratio $k:1$. Then, the position vector of the wicketkeeper $\vec{W}$ can be expressed as: $$ \vec{W} = \frac{k\vec{F} + 1\vec{B}}{k+1} $$

Step 3: Substituting the Given Vectors

We are given $\vec{B}=2\hat{i}+8\hat{j}$, $\vec{W}=6\hat{i}+12\hat{j}$ and $\vec{F}=12\hat{i}+18\hat{j}$. Substituting these into the section formula: $$ 6\hat{i}+12\hat{j} = \frac{k(12\hat{i}+18\hat{j}) + (2\hat{i}+8\hat{j})}{k+1} $$

Step 4: Equating the Components

Equating the $\hat{i}$ and $\hat{j}$ components, we get: $$ 6 = \frac{12k + 2}{k+1} \quad \text{and} \quad 12 = \frac{18k + 8}{k+1} $$

Step 5: Solving for k

From the first equation: $$ 6(k+1) = 12k + 2 \\ 6k + 6 = 12k + 2 \\ 6k = 4 \\ k = \frac{4}{6} = \frac{2}{3} $$ From the second equation: $$ 12(k+1) = 18k + 8 \\ 12k + 12 = 18k + 8 \\ 6k = 4 \\ k = \frac{4}{6} = \frac{2}{3} $$ Both equations give the same value for $k$.

Step 6: Finding the Ratio

The ratio is $k:1$, which is $\frac{2}{3}:1$. Multiplying by 3 to get rid of the fraction, we get the ratio $2:3$.

Final Answer: 2:3

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the section formula to solve the problem in a real-world context (cricket match).
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding and application of the section formula in vector algebra, which is a conceptual understanding.<\/span>
Syllabus Audit: In the context of CBSE Class 12, this is classified as APPLICATION. The question applies the concept of vectors and the section formula to a practical scenario.

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