Class CBSE Class 12 Mathematics Three Dimensional Geometry Q #914
KNOWLEDGE BASED
APPLY
2 Marks 2023 VSA
Position vectors of the points A, B and C as shown in the figure below are a, $\vec{b}$ and $\vec{c}$ respectively. If $\vec{AC}=\frac{5}{4}\vec{AB}$ , express $\vec{c}$ in terms of $\vec{a}$ and $\vec{b}$ .
OR Check whether the lines given by equations $x=2\lambda+2$, $y=7\lambda+1$, $z=-3\lambda-3$ and $x=-\mu-2,$ $y=2\mu+8,$ $z=4\mu+5$ are perpendicular to each other or not.
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  1. Question 1:

    Given: $\vec{AC} = \frac{5}{4}\vec{AB}$

    We know that $\vec{AC} = \vec{c} - \vec{a}$ and $\vec{AB} = \vec{b} - \vec{a}$

    Substituting these into the given equation, we get: $\vec{c} - \vec{a} = \frac{5}{4}(\vec{b} - \vec{a})$

    Multiplying out the right side: $\vec{c} - \vec{a} = \frac{5}{4}\vec{b} - \frac{5}{4}\vec{a}$

    Isolating $\vec{c}$: $\vec{c} = \vec{a} + \frac{5}{4}\vec{b} - \frac{5}{4}\vec{a}$

    Combining terms: $\vec{c} = -\frac{1}{4}\vec{a} + \frac{5}{4}\vec{b}$

  2. Question 2:

    The direction ratios of the first line are (2, 7, -3).

    The direction ratios of the second line are (-1, 2, 4).

    For the lines to be perpendicular, the dot product of their direction ratios must be zero.

    So, (2)(-1) + (7)(2) + (-3)(4) = -2 + 14 - 12 = 0.

    Since the dot product is zero, the lines are perpendicular to each other.

Correct Answer: Question 1: $\vec{c} = -\frac{1}{4}\vec{a} + \frac{5}{4}\vec{b}$ , Question 2: Yes, the lines are perpendicular.

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of vectors and position vectors to express one vector in terms of others, given a specific relationship between them.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to manipulate vector equations and express one vector in terms of others. This involves algebraic manipulation and application of vector properties.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding and application of vector algebra concepts as taught in the textbook.

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