Class CBSE Class 12 Mathematics Three Dimensional Geometry Q #1824
COMPETENCY BASED
APPLY
5 Marks 2026 AISSCE(Board Exam) LA
Represent the equations of lines $l_{1}$ and $l_{2}$ in vector form and check whether they are intersecting or not. $l_{1}:\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$, $l_{2}:\frac{x+1}{-1}=\frac{2-y}{-2}=\frac{z-5}{5}$
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Detailed Solution

Step 1: Convert lines to standard vector form

The standard form of a line is (x-x1)/a = (y-y1)/b = (z-z1)/c. For l1: (x+3)/-3 = (y-1)/1 = (z-5)/5. The point is A(-3, 1, 5) and direction vector is b1 = -3i + j + 5k. Vector form: r = (-3i + j + 5k) + λ(-3i + j + 5k).

Step 2: Standardize l2

Rewrite l2: (x+1)/-1 = (y-2)/2 = (z-5)/5. The point is B(-1, 2, 5) and direction vector is b2 = -i + 2j + 5k. Vector form: r = (-i + 2j + 5k) + μ(-i + 2j + 5k).

Step 3: Check for intersection

Two lines intersect if the shortest distance is zero, or if there exist λ and μ such that the coordinates are equal. Equating components: $$-3 - 3λ = -1 - μ$$ $$1 + λ = 2 + 2μ$$ $$5 + 5λ = 5 + 5μ$$ From the third equation, λ = μ. Substituting into the second: 1 + λ = 2 + 2λ, which gives λ = -1. Checking the first: -3 - 3(-1) = 0 and -1 - (-1) = 0. Since 0 = 0, the lines intersect.

Final Answer: The lines intersect at the point (0, 0, 0).

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must transform Cartesian equations into vector form and apply the condition of consistency for linear systems.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a specific algorithmic sequence to convert forms and solve a system of linear equations to determine intersection.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the ability to manipulate 3D geometry representations beyond rote memorization.

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