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Cheat sheet · Class 12 · CBSE · Commerce Mathematics

Three Dimensional Geometry cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

CBSE Class 12 · Mathematics

Cheat sheet

Chapter 11Three Dimensional Geometry

Skew lines: shortest distance = |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|.

1. Direction Cosines

  • l = cos α, m = cos β, n = cos γ.
  • Direction ratios a, b, c are proportional to l, m, n.

2. Lines

r⃗=a⃗+λb⃗\vec r = \vec a + \lambda\vec bx−x1a=y−y1b=z−z1c\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}
cos⁡θ=∣b⃗1⋅b⃗2∣b⃗1∣∣b⃗2∣∣\cos\theta = \left|\dfrac{\vec b_1\cdot\vec b_2}{|\vec b_1||\vec b_2|}\right|
d=∣(a⃗2−a⃗1)⋅(b⃗1×b⃗2)∣b⃗1×b⃗2∣∣d = \left|\dfrac{(\vec a_2 - \vec a_1)\cdot(\vec b_1\times\vec b_2)}{|\vec b_1\times\vec b_2|}\right|

Parallel lines: d=∣b⃗×(a⃗2−a⃗1)∣∣b⃗∣d = \dfrac{|\vec b\times(\vec a_2 - \vec a_1)|}{|\vec b|}.

3. Key Points

  • Skew lines: neither parallel nor intersecting.
  • Perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0.
  • Cartesian ↔ vector form conversion.
  • Shortest distance 0 ⇒ lines intersect.

Remember: DCs & DRs | line equations | angle | shortest distance

Three Dimensional Geometry: questions and notes

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