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 $\hat{i}+\hat{j}+\hat{k}.$
Prev Next

AI Tutor Explanation

Powered by Gemini

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 $-3\hat{i} - 3\hat{j} - 3\hat{k}$

AI generated content. Review strictly for academic accuracy.

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.

More from this Chapter

MCQ_SINGLE
If \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), \(|\vec{a}| = \sqrt{37}\), \(|\vec{b}| = 3\) and \(|\vec{c}| = 4\), then the angle between \(\vec{b}\) and \(\vec{c}\) is
VSA
If $\vec{r}=3\hat{i}-2\hat{j}+6\hat{k}$, find the value of $(\vec{r}\times\hat{j})\cdot(\vec{r}\times\hat{k})-12$
MCQ_SINGLE
If \vec{a}+\vec{b}=\hat{i} and \vec{a}=2\hat{i}-2\hat{j}+2\hat{k}, then |\vec{b}| equals:
MCQ_SINGLE
If the sides AB and AC of \(\triangle ABC\) are represented by vectors \(\hat{j}+\hat{k}\) and \(3\hat{i}-\hat{j}+4\hat{k}\) respectively, then the length of the median through A on BC is:
MCQ_SINGLE
If a vector makes an angle of $\frac{\pi}{4}$ with the positive directions of both x-axis and y-axis, then the angle which it makes with positive z-axis is :
View All Questions