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11Class 12 Maths

Three Dimensional Geometry formulas

Direction cosines and line formulas for Three Dimensional Geometry. Most problems use direction ratios, so get comfortable with them.

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All Maths formulas

Direction cosines

Relation
l2+m2+n2=1\displaystyle l^2 + m^2 + n^2 = 1
From direction ratios
l=aa2+b2+c2\displaystyle l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}

Similarly for m and n.

Through two points
x2−x1, y2−y1, z2−z1\displaystyle x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1

These are the direction ratios.

Lines

Vector form
r⃗=a⃗+λb⃗\displaystyle \vec r = \vec a + \lambda\vec b
Cartesian form
x−x1a=y−y1b=z−z1c\displaystyle \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}
Angle between lines
cos⁡θ=∣a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22∣\displaystyle \cos\theta = \left|\frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}\right|
Perpendicular lines
a1a2+b1b2+c1c2=0\displaystyle a_1a_2 + b_1b_2 + c_1c_2 = 0

Shortest distance

Skew lines
d=∣(b⃗1×b⃗2)⋅(a⃗2−a⃗1)∣b⃗1×b⃗2∣∣\displaystyle d = \left|\frac{(\vec b_1\times\vec b_2)\cdot(\vec a_2 - \vec a_1)}{|\vec b_1\times\vec b_2|}\right|
Parallel lines
d=∣b⃗×(a⃗2−a⃗1)∣∣b⃗∣\displaystyle d = \frac{|\vec b\times(\vec a_2 - \vec a_1)|}{|\vec b|}

Where students slip up

  • Treating direction ratios as direction cosines. Only cosines satisfy l² + m² + n² = 1.
  • Using the skew-line formula for parallel lines, where b₁ × b₂ = 0 makes it undefined.

Try it on your weakest chapter.

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