Class CBSE Class 12 Mathematics Vector Algebra Q #876
KNOWLEDGE BASED
APPLY
2 Marks 2023 VSA
Find all the vectors of magnitude $3\sqrt{3}$ which are collinear to vector $\hat{i}+\hat{j}+\hat{k}.$
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Step-by-Step Solution

  1. Let the given vector be $\vec{a} = \hat{i} + \hat{j} + \hat{k}$.

  2. Find the magnitude of $\vec{a}$: $|\vec{a}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$.

  3. Find the unit vector along $\vec{a}$: $\hat{a} = \frac{\vec{a}}{|\vec{a}|} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} = \frac{1}{\sqrt{3}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \frac{1}{\sqrt{3}}\hat{k}$.

  4. We need to find vectors of magnitude $3\sqrt{3}$ which are collinear to $\vec{a}$. These vectors will be of the form $\pm 3\sqrt{3} \hat{a}$.

  5. So, the required vectors are $\pm 3\sqrt{3} \left( \frac{1}{\sqrt{3}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \frac{1}{\sqrt{3}}\hat{k} \right) = \pm (3\hat{i} + 3\hat{j} + 3\hat{k})$.

  6. Therefore, the vectors are $3\hat{i} + 3\hat{j} + 3\hat{k}$ and $-3\hat{i} - 3\hat{j} - 3\hat{k}$.

Correct Answer: $3\hat{i} + 3\hat{j} + 3\hat{k}$ and $-3\hat{i} - 3\hat{j} - 3\hat{k}$

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of vectors, magnitude, and collinearity to find the required vectors. They must use the given information to construct the solution.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a specific procedure to be followed: finding the unit vector, scaling it to the desired magnitude, and considering both positive and negative directions.
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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