Class CBSE Class 12 Mathematics Vector Algebra Q #1426
KNOWLEDGE BASED
UNDERSTAND
2 Marks 2025 AISSCE(Board Exam) VSA
Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three vectors such that $\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}$ and $\vec{a}\times\vec{b}=\vec{a}\times\vec{c}$, $\vec{a}\ne\vec{0}$. Show that $\vec{b}=\vec{c}$.
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Detailed Solution

Step 1: Use the given dot product condition

We are given that $\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}$. This can be rewritten as $\vec{a}\cdot\vec{b} - \vec{a}\cdot\vec{c} = 0$, which simplifies to $\vec{a}\cdot(\vec{b}-\vec{c}) = 0$.

Step 2: Use the given cross product condition

We are given that $\vec{a}\times\vec{b}=\vec{a}\times\vec{c}$. This can be rewritten as $\vec{a}\times\vec{b} - \vec{a}\times\vec{c} = \vec{0}$, which simplifies to $\vec{a}\times(\vec{b}-\vec{c}) = \vec{0}$.

Step 3: Analyze the implications of the dot and cross products

From $\vec{a}\cdot(\vec{b}-\vec{c}) = 0$, we know that $\vec{a}$ is perpendicular to $(\vec{b}-\vec{c})$. From $\vec{a}\times(\vec{b}-\vec{c}) = \vec{0}$, we know that $\vec{a}$ is parallel to $(\vec{b}-\vec{c})$.

Step 4: Combine the information to conclude

Since $\vec{a}$ is both perpendicular and parallel to $(\vec{b}-\vec{c})$, and $\vec{a} \ne \vec{0}$, the only possibility is that $(\vec{b}-\vec{c}) = \vec{0}$. Therefore, $\vec{b} = \vec{c}$.

Final Answer: $\vec{b}=\vec{c}$

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to comprehend the properties of dot and cross products to deduce the relationship between the vectors.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of dot product and cross product, and their geometric interpretations (perpendicularity and parallelism).
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of vector algebra, specifically the properties of dot and cross products as covered in the textbook.

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