Q1448: Vector $\vec{r}$ is inclined at equal angles to the three ax...

Class CBSE Class 12 Mathematics Vector Algebra Q #1448

KNOWLEDGE BASED

REMEMBER

2 Marks 2025 AISSCE(Board Exam) VSA

Vector $\vec{r}$ is inclined at equal angles to the three axes x, y and z. If magnitude of $\vec{r}$ is $5\sqrt{3}$ units, then find $\vec{r}$.

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AI Tutor Explanation

Detailed Solution

Step 1: Define direction cosines

Let the vector $\vec{r}$ be inclined at an angle $\alpha$ with each of the x, y, and z axes. Then, the direction cosines of $\vec{r}$ are $l = \cos\alpha$, $m = \cos\alpha$, and $n = \cos\alpha$.

Step 2: Use the property of direction cosines

We know that $l^2 + m^2 + n^2 = 1$. Substituting the values of $l, m, n$, we get: $$(\cos\alpha)^2 + (\cos\alpha)^2 + (\cos\alpha)^2 = 1$$ $$3\cos^2\alpha = 1$$ $$\cos^2\alpha = \frac{1}{3}$$ $$\cos\alpha = \pm\frac{1}{\sqrt{3}}$$

Step 3: Find the direction cosines

Thus, the direction cosines are $l = \pm\frac{1}{\sqrt{3}}$, $m = \pm\frac{1}{\sqrt{3}}$, and $n = \pm\frac{1}{\sqrt{3}}$.

Step 4: Express the vector in terms of its magnitude and direction cosines

Given that the magnitude of $\vec{r}$ is $|\vec{r}| = 5\sqrt{3}$. The vector $\vec{r}$ can be written as: $$\vec{r} = |\vec{r}|(l\hat{i} + m\hat{j} + n\hat{k})$$ $$\vec{r} = 5\sqrt{3}\left(\pm\frac{1}{\sqrt{3}}\hat{i} \pm \frac{1}{\sqrt{3}}\hat{j} \pm \frac{1}{\sqrt{3}}\hat{k}\right)$$ $$\vec{r} = \pm 5\hat{i} \pm 5\hat{j} \pm 5\hat{k}$$

Step 5: Write the final vector

Therefore, $\vec{r} = \pm 5(\hat{i} + \hat{j} + \hat{k})$.

Final Answer: $\vec{r} = \pm 5(\hat{i} + \hat{j} + \hat{k})$

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because the student needs to recall the formula for direction cosines and their relationship to the magnitude of a vector.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concept of direction cosines and how they relate to a vector's components.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of vector representation and direction cosines, which are core concepts in the syllabus.