Class CBSE Class 12 Mathematics Vector Algebra Q #1479
KNOWLEDGE BASED
REMEMBER
3 Marks 2025 AISSCE(Board Exam) SA
The scalar product of the vector $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$ with a unit vector along sum of vectors $\vec{b}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{c}=\lambda\hat{i}-2\hat{j}-3\hat{k}$ is equal to 1. Find the value of $\lambda$.
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Detailed Solution

Step 1: Find the sum of vectors $\vec{b}$ and $\vec{c}$

We are given $\vec{b}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{c}=\lambda\hat{i}-2\hat{j}-3\hat{k}$. So, $\vec{b}+\vec{c} = (2+\lambda)\hat{i} + (-4-2)\hat{j} + (5-3)\hat{k} = (2+\lambda)\hat{i} -6\hat{j} + 2\hat{k}$.

Step 2: Find the unit vector along $\vec{b}+\vec{c}$

Let $\hat{u}$ be the unit vector along $\vec{b}+\vec{c}$. Then, $\hat{u} = \frac{\vec{b}+\vec{c}}{|\vec{b}+\vec{c}|}$. First, we find the magnitude of $\vec{b}+\vec{c}$: $|\vec{b}+\vec{c}| = \sqrt{(2+\lambda)^2 + (-6)^2 + (2)^2} = \sqrt{(2+\lambda)^2 + 36 + 4} = \sqrt{(2+\lambda)^2 + 40}$. So, $\hat{u} = \frac{(2+\lambda)\hat{i} -6\hat{j} + 2\hat{k}}{\sqrt{(2+\lambda)^2 + 40}}$.

Step 3: Find the scalar product of $\vec{a}$ and $\hat{u}$

We are given $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$. The scalar product of $\vec{a}$ and $\hat{u}$ is given by $\vec{a} \cdot \hat{u} = 1$. So, $(\hat{i}-\hat{j}+2\hat{k}) \cdot \frac{(2+\lambda)\hat{i} -6\hat{j} + 2\hat{k}}{\sqrt{(2+\lambda)^2 + 40}} = 1$. $\frac{(1)(2+\lambda) + (-1)(-6) + (2)(2)}{\sqrt{(2+\lambda)^2 + 40}} = 1$. $\frac{2+\lambda + 6 + 4}{\sqrt{(2+\lambda)^2 + 40}} = 1$. $\frac{\lambda + 12}{\sqrt{(2+\lambda)^2 + 40}} = 1$.

Step 4: Solve for $\lambda$

Squaring both sides, we get: $(\lambda + 12)^2 = (2+\lambda)^2 + 40$. $\lambda^2 + 24\lambda + 144 = \lambda^2 + 4\lambda + 4 + 40$. $24\lambda + 144 = 4\lambda + 44$. $20\lambda = -100$. $\lambda = -5$.

Final Answer: -5

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because the student needs to recall the formula for the scalar product of two vectors and the definition of a unit vector to solve the problem.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a series of steps, including finding the sum of vectors, calculating the unit vector, and using the scalar product formula.
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 concepts as covered in the textbook.

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