Q1447: If $\vec{a}$ and $\vec{b}$ are position vectors of point A a...

Class CBSE Class 12 Mathematics Vector Algebra Q #1447

KNOWLEDGE BASED

REMEMBER

2 Marks 2025 AISSCE(Board Exam) VSA

If $\vec{a}$ and $\vec{b}$ are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that $BC=3BA$.

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AI Tutor Explanation

Detailed Solution

Step 1: Define the vectors

Let $\vec{a}$ be the position vector of point A and $\vec{b}$ be the position vector of point B. Let $\vec{c}$ be the position vector of point C.

Step 2: Express $\vec{BC}$ and $\vec{BA}$ in terms of position vectors

We have $\vec{BC} = \vec{c} - \vec{b}$ and $\vec{BA} = \vec{a} - \vec{b}$.

Step 3: Use the given condition $BC = 3BA$

Since $BC = 3BA$, we can write $\vec{BC} = 3\vec{BA}$. Substituting the expressions from Step 2, we get: $$ \vec{c} - \vec{b} = 3(\vec{a} - \vec{b}) $$

Step 4: Solve for $\vec{c}$

Now, we solve for $\vec{c}$: $$ \vec{c} = \vec{b} + 3(\vec{a} - \vec{b}) $$ $$ \vec{c} = \vec{b} + 3\vec{a} - 3\vec{b} $$ $$ \vec{c} = 3\vec{a} - 2\vec{b} $$

Final Answer: $3\vec{a} - 2\vec{b}$

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because it requires recalling the section formula and applying it directly.

Knowledge Dimension: CONCEPTUAL

Justification: The question tests understanding of position vectors and their relationships in geometric contexts, specifically the concept of collinearity and ratios.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly relates to the application of vector algebra in 3D geometry, a core concept in the syllabus.