Class CBSE Class 12 Mathematics Vector Algebra Q #1360
COMPETENCY BASED
REMEMBER
2 Marks 2025 AISSCE(Board Exam) VSA
Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors $\vec{a}=3\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+4\hat{k}$. Determine the angle formed between the kite strings. Assume there is no slack in the strings.
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Detailed Solution

Step 1: Recall the formula for the angle between two vectors

The angle $\theta$ between two vectors $\vec{a}$ and $\vec{b}$ is given by: $$cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$$

Step 2: Calculate the dot product of the vectors

Given $\vec{a}=3\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+4\hat{k}$, the dot product is: $$\vec{a} \cdot \vec{b} = (3)(2) + (1)(-2) + (2)(4) = 6 - 2 + 8 = 12$$

Step 3: Calculate the magnitude of vector $\vec{a}$

The magnitude of $\vec{a}$ is: $$|\vec{a}| = \sqrt{(3)^2 + (1)^2 + (2)^2} = \sqrt{9 + 1 + 4} = \sqrt{14}$$

Step 4: Calculate the magnitude of vector $\vec{b}$

The magnitude of $\vec{b}$ is: $$|\vec{b}| = \sqrt{(2)^2 + (-2)^2 + (4)^2} = \sqrt{4 + 4 + 16} = \sqrt{24} = 2\sqrt{6}$$

Step 5: Calculate the cosine of the angle

$$cos(\theta) = \frac{12}{\sqrt{14} \cdot 2\sqrt{6}} = \frac{6}{\sqrt{14 \cdot 6}} = \frac{6}{\sqrt{84}} = \frac{6}{\sqrt{4 \cdot 21}} = \frac{6}{2\sqrt{21}} = \frac{3}{\sqrt{21}}$$ Rationalizing the denominator: $$cos(\theta) = \frac{3\sqrt{21}}{21} = \frac{\sqrt{21}}{7}$$

Step 6: Find the angle

$$\theta = cos^{-1}(\frac{\sqrt{21}}{7})$$

Final Answer: $cos^{-1}(\frac{\sqrt{21}}{7})$

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because it requires recalling the formula for the angle between two vectors and applying it to the given problem.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concept of vectors and the formula to calculate the angle between them. It involves applying the formula rather than just recalling a fact.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It is competency based because NATURE_REASONING requires application of vector algebra to a real-world scenario (kite flying), testing the student's ability to apply learned concepts.

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