Class CBSE Class 12 Mathematics Vector Algebra Q #552
COMPETENCY BASED
APPLY
1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE
Let \(\vec{a}\) be any vector such that \(|\vec{a}|=a\) The value of \(|\vec{a}\times\hat{i}|^{2}+|\vec{a}\times\hat{j}|^{2}+|\vec{a}\times\hat{k}|^{2}\) is:
(A) \(a^{2}\)
(B) \(2a^{2}\)
(C) \(3a^{2}\)
(D) 0
Prev Next

AI Tutor Explanation

Powered by Gemini

Step-by-Step Solution

Let \(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\). Then \(|\vec{a}|^2 = a_1^2 + a_2^2 + a_3^2 = a^2\).

Now, we compute the cross products:

\(\vec{a} \times \hat{i} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) \times \hat{i} = a_2(\hat{j} \times \hat{i}) + a_3(\hat{k} \times \hat{i}) = -a_2\hat{k} + a_3\hat{j}\)

\(|\vec{a} \times \hat{i}|^2 = a_2^2 + a_3^2\)

\(\vec{a} \times \hat{j} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) \times \hat{j} = a_1(\hat{i} \times \hat{j}) + a_3(\hat{k} \times \hat{j}) = a_1\hat{k} - a_3\hat{i}\)

\(|\vec{a} \times \hat{j}|^2 = a_1^2 + a_3^2\)

\(\vec{a} \times \hat{k} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) \times \hat{k} = a_1(\hat{i} \times \hat{k}) + a_2(\hat{j} \times \hat{k}) = -a_1\hat{j} + a_2\hat{i}\)

\(|\vec{a} \times \hat{k}|^2 = a_1^2 + a_2^2\)

Adding these squared magnitudes:

\(|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2 = (a_2^2 + a_3^2) + (a_1^2 + a_3^2) + (a_1^2 + a_2^2) = 2(a_1^2 + a_2^2 + a_3^2) = 2a^2\)

Correct Answer: 2a^{2}

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 cross product and magnitude of vectors to solve the problem.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding of vector algebra, cross products, and their properties.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply vector algebra concepts to solve a problem, aligning with competency-based assessment.

More from this Chapter

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 $\lambda$.
VSA
If $\vec{\alpha}$ and $\vec{\beta}$ are position vectors of two points P and Q respectively, then find the position vector of a point R in QP produced such that $QR=\frac{3}{2}QP$.
VSA
(a) If the vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a}| = 3$, $|\vec{b}| = \frac{2}{3}$ and $\vec{a} \times \vec{b}$ is a unit vector, then find the angle between $\vec{a}$ and $\vec{b}$. OR(b) Find the area of a parallelogram whose adjacent sides are determined by the vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$.
MCQ_SINGLE
The vectors \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\), \(\vec{b}=\hat{i}-3\hat{j}-5\hat{k}\) and \(\vec{c}=-3\hat{i}+4\hat{j}+4\hat{k}\) represents the sides of
SUBJECTIVE
(i) Complete the given figure to explain their entire movement plan along the respective vectors. (ii) Find vectors $\vec{AC}$ and $\vec{BC}$. (iii) (a) If $\vec{a} \cdot \vec{b} = 1$, distance of O to A is 1 km and that from O to B is 2 km, then find the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$. Also, find $| \vec{a} \times \vec{b}|$. OR (iii) (b) If $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$, then find a unit vector perpendicular to $(\vec{a}+\vec{b})$ and $(\vec{a}-\vec{b})$.
View All Questions