Three Dimensional Geometry Quiz 1

 

Three Dimensional Geometry – Interactive Quiz 1

Topic: Direction Cosines & Direction Ratios
(Worked examples converted into quiz format)

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Key Formulae
Direction cosines: l = cosα, m = cosβ, n = cosγ
Condition: l² + m² + n² = 1
If direction ratios are (a, b, c):
l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²)

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Q1. If the direction ratios of a line are (2, −2, 1), the direction cosine along x-axis is:

2/3
2/√9
2/√6
√6/2

Solution:
√(2²+(-2)²+1²)=√9=3
l = 2/3

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Q2. If a line makes equal angles with all three coordinate axes, find its direction cosine (nearest 3 decimals).

Solution:
l = m = n
3l² = 1 ⇒ l = 1/√3 ≈ 0.577

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Q3. The direction ratios of a line perpendicular to the plane x + 2y − 2z + 5 = 0 are:

(1, −2, 2)
(2, 1, −2)
(1, 2, −2)
(−1, 2, 2)

Solution:
Normal to plane = coefficients of x,y,z = (1,2,−2)

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Q4. If the direction cosines of a line are (l, m, n) and l = m, find the value of n².

Solution:
2l² + n² = 1
If l = m = 0 ⇒ n² = 1

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