Class CBSE Class 12 Mathematics Vector Algebra Q #1358
KNOWLEDGE BASED
REMEMBER
2 Marks 2025 AISSCE(Board Exam) VSA
The diagonals of a parallelogram are given by $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$. Find the area of the parallelogram.
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Detailed Solution

Step 1: Recall the formula for the area of a parallelogram in terms of its diagonals.

The area of a parallelogram whose diagonals are given by vectors $\vec{a}$ and $\vec{b}$ is given by half the magnitude of the cross product of the diagonals: $$Area = \frac{1}{2} |\vec{a} \times \vec{b}|$$

Step 2: Compute the cross product $\vec{a} \times \vec{b}$.

Given $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$, we compute the cross product as follows: $$ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 1 \\ 1 & 3 & -1 \end{vmatrix} $$ $$ = \hat{i}((-1)(-1) - (1)(3)) - \hat{j}((2)(-1) - (1)(1)) + \hat{k}((2)(3) - (-1)(1)) $$ $$ = \hat{i}(1 - 3) - \hat{j}(-2 - 1) + \hat{k}(6 + 1) $$ $$ = -2\hat{i} + 3\hat{j} + 7\hat{k} $$

Step 3: Calculate the magnitude of the cross product.

The magnitude of $\vec{a} \times \vec{b} = -2\hat{i} + 3\hat{j} + 7\hat{k}$ is: $$ |\vec{a} \times \vec{b}| = \sqrt{(-2)^2 + (3)^2 + (7)^2} = \sqrt{4 + 9 + 49} = \sqrt{62} $$

Step 4: Calculate the area of the parallelogram.

The area of the parallelogram is half the magnitude of the cross product: $$ Area = \frac{1}{2} |\vec{a} \times \vec{b}| = \frac{1}{2} \sqrt{62} $$

Final Answer: $\frac{\sqrt{62}}{2}$ square units

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because the student needs to recall the formula for the area of a parallelogram given its diagonals and apply it.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific procedure (calculating the cross product and its magnitude) to find the area.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of vector algebra and its applications in geometry, specifically finding the area of a parallelogram using diagonals.

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