Chapter 11: Three Dimensional Geometry
Key Formulas:
• Direction cosines: l² + m² + n² = 1
• Line: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c
• Distance from point (x₁,y₁,z₁) to plane ax+by+cz+d=0:
d = |ax₁+by₁+cz₁+d|/√(a²+b²+c²)
Q1. Find the direction cosines of the line making equal angles with the coordinate axes.
Solution
Let l = m = n (equal angles)
l² + m² + n² = 1 ⇒ 3l² = 1
l = m = n = 1/√3
Q2. Find the equation of the line passing through (1, 2, 3) and parallel to (2, 3, 4).
Solution
(x − 1)/2 = (y − 2)/3 = (z − 3)/4
Q3. Find the shortest distance between the lines: r = ⋂⃗i + ⋂⃗j + λ(⋂⃗i + ⋂⃗j + ⋂⃗k) and r = 2⋂⃗i + ⋂⃗j + μ(⋂⃗i + ⋂⃗j − 2⋂⃗k).
Solution
a₁ = ⋂⃗i+⋂⃗j, a₂ = 2⋂⃗i+⋂⃗j, b₁ = ⋂⃗i+⋂⃗j+⋂⃗k, b₂ = ⋂⃗i+⋂⃗j−2⋂⃗k
a₂ − a₁ = ⋂⃗i
b₁ × b₂ = |⋂⃗i ⋂⃗j ⋂⃗k; 1 1 1; 1 1 −2| = ⋂⃗i(−2−1) − ⋂⃗j(−2−1) + ⋂⃗k(1−1) = −3⋂⃗i+3⋂⃗j
|(a₂−a₁) · (b₁×b₂)| = |−3+0| = 3
|b₁×b₂| = √(9+9) = 3√2
d = 3/(3√2) = 1/√2
Q4. Find the equation of the plane passing through (1, 1, 1) and perpendicular to x − 2y + 3z = 7.
Solution
Normal to the given plane = (1, −2, 3)
Required plane has same normal direction:
1(x−1) − 2(y−1) + 3(z−1) = 0
x − 2y + 3z − 1 + 2 − 3 = 0
x − 2y + 3z − 2 = 0
Q5. Find the distance of the point (3, −2, 1) from the plane 2x − y + 2z + 3 = 0.
Solution
d = |2(3) − 1(−2) + 2(1) + 3|/√(4+1+4)
= |6 + 2 + 2 + 3|/√9
= 13/3 = 13/3 units