Class CBSE Class 12 Mathematics Vector Algebra Q #1294
COMPETENCY BASED
UNDERSTAND
2 Marks 2024 AISSCE(Board Exam) VSA
In the given figure, ABCD is a parallelogram. If $\vec{AB}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{DB}=3\hat{i}-6\hat{j}+2\hat{k}$ , then find $\vec{AD}$ and hence find the area of parallelogram ABCD.
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Detailed Solution

Step 1: Find $\vec{AD}$ using the triangle law of vector addition.

In triangle ABD, we have $\vec{AD} + \vec{DB} = \vec{AB}$. Therefore, $\vec{AD} = \vec{AB} - \vec{DB}$.

Step 2: Calculate $\vec{AD}$.

Given $\vec{AB}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{DB}=3\hat{i}-6\hat{j}+2\hat{k}$, we can find $\vec{AD}$ as follows: $$ \vec{AD} = (2\hat{i}-4\hat{j}+5\hat{k}) - (3\hat{i}-6\hat{j}+2\hat{k}) = (2-3)\hat{i} + (-4+6)\hat{j} + (5-2)\hat{k} = -\hat{i} + 2\hat{j} + 3\hat{k} $$

Step 3: Find the area of parallelogram ABCD.

The area of parallelogram ABCD is given by the magnitude of the cross product of $\vec{AB}$ and $\vec{AD}$. $$ \text{Area} = |\vec{AB} \times \vec{AD}| $$

Step 4: Calculate the cross product $\vec{AB} \times \vec{AD}$.

$$ \vec{AB} \times \vec{AD} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -4 & 5 \\ -1 & 2 & 3 \end{vmatrix} = \hat{i}((-4)(3) - (5)(2)) - \hat{j}((2)(3) - (5)(-1)) + \hat{k}((2)(2) - (-4)(-1)) $$ $$ = \hat{i}(-12 - 10) - \hat{j}(6 + 5) + \hat{k}(4 - 4) = -22\hat{i} - 11\hat{j} + 0\hat{k} $$

Step 5: Calculate the magnitude of $\vec{AB} \times \vec{AD}$.

$$ |\vec{AB} \times \vec{AD}| = \sqrt{(-22)^2 + (-11)^2 + 0^2} = \sqrt{484 + 121} = \sqrt{605} = 11\sqrt{5} $$

Final Answer: $\vec{AD} = -\hat{i} + 2\hat{j} + 3\hat{k}$ and Area = $11\sqrt{5}$ square units

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of vector addition and cross product to find the unknown vector and the area of the parallelogram.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding of vector addition, cross product, and their geometric interpretations (area of a parallelogram). It's not just recall of formulas but application of concepts.
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 geometric problem, which aligns with competency-based assessment.

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