Three Dimensional Geometry – Quiz 2

Topic: Equation of Line in 3D
(Worked examples converted into interactive quiz)

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Key Formulae
Vector form: r = a + λb
Cartesian form: (x − x₁)/l = (y − y₁)/m = (z − z₁)/n
Distance between two points:
√[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]

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Q1. The direction ratios of the line passing through (1,2,3) and (3,6,7) are:

(2,2,2)
(1,2,2)
(2,4,4)
(3,6,7)

Solution:
Direction ratios = (3−1,6−2,7−3) = (2,4,4)

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Q2. Which of the following is the Cartesian equation of a line passing through (2,−1,3) with direction ratios (1,2,−1)?

(x−2)/2 = (y+1)/1 = (z−3)/(−1)
(x−2)/1 = (y+1)/2 = (z−3)/(−1)
(x+2)/1 = (y−1)/2 = (z+3)/(−1)
(x−2)/1 = (y−1)/2 = (z+3)

Solution:
Use (x−x₁)/l = (y−y₁)/m = (z−z₁)/n

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Q3. Find the distance between points A(1,2,3) and B(4,6,6).

Solution:
Distance = √[(3)²+(4)²+(3)²] = √34 ≈ 5.83

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Q4. The vector equation of a line passing through the origin and parallel to vector 2i − j + 3k is:

r = λ(2i − j + 3k)
r = i + λ(2i − j + 3k)
r = (2i − j + 3k) + λ
r = λ(i + j + k)

Solution:
Line through origin: r = λb

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Q5. If the direction ratios of a line are proportional to (2, −2, 1), find the value of l²+m²+n².

Solution:
For direction cosines: l²+m²+n² = 1

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