The Chapter in One Sweep

Directions

A line's direction is a triple: direction cosines (l,m,n)=(cos⁡α,cos⁡β,cos⁡γ)(l, m, n) = (\cos\alpha, \cos\beta, \cos\gamma) with l2+m2+n2=1l^2 + m^2 + n^2 = 1 (two sets per undirected line, differing by an overall sign), or any proportional direction ratios (a,b,c)(a, b, c) — normalise by a2+b2+c2\sqrt{a^2 + b^2 + c^2} to recover cosines. Through two points, the coordinate differences are ratios, and dividing by the distance gives cosines. Collinearity of points = proportional ratios of consecutive segments.

Directed line with direction angles and the ratios to cosines pipeline

The line, both ways

r⃗=a⃗+λb⃗⟺x−x1a=y−y1b=z−z1c\vec{r} = \vec{a} + \lambda\vec{b} \qquad\Longleftrightarrow\qquad \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}

Point from a⃗\vec{a} (numerators), direction from b⃗\vec{b} (denominators). Two-point form: direction b⃗−a⃗\vec{b} - \vec{a}. Standardise before reading: numerators x−x1x - x_1, unit coefficients.

Line through point A parallel to vector b with position vector r

Angle between lines

cos⁡θ=∣a1a2+b1b2+c1c2∣a12+b12+c12a22+b22+c22\cos\theta = \frac{\left|a_1a_2 + b_1b_2 + c_1c_2\right|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}

Perpendicular: a1a2+b1b2+c1c2=0a_1a_2 + b_1b_2 + c_1c_2 = 0. Parallel: proportional ratios. With direction cosines, the denominators vanish.

Shortest distance

skew: d=∣(b⃗1×b⃗2)⋅(a⃗2−a⃗1)∣∣b⃗1×b⃗2∣parallel: d=∣b⃗×(a⃗2−a⃗1)∣∣b⃗∣\text{skew: } d = \frac{\left|\left(\vec{b}_1\times\vec{b}_2\right)\cdot\left(\vec{a}_2-\vec{a}_1\right)\right|}{\left|\vec{b}_1\times\vec{b}_2\right|} \qquad \text{parallel: } d = \frac{\left|\vec{b}\times\left(\vec{a}_2-\vec{a}_1\right)\right|}{\left|\vec{b}\right|}

d=0d = 0 for non-parallel lines means they intersect (the numerator is the coplanarity triple product).

Two skew lines with the common perpendicular segment and both formulas

The JEE Plane Sheet, and the Mistake Checklist

Planes (JEE retains what NCERT dropped)

Item Formula
General form ax+by+cz+d=0ax + by + cz + d = 0, normal ratios (a,b,c)(a, b, c)
Point-normal a(x−x1)+b(y−y1)+c(z−z1)=0a(x - x_1) + b(y - y_1) + c(z - z_1) = 0
Intercept form xp+yq+zr=1\dfrac{x}{p} + \dfrac{y}{q} + \dfrac{z}{r} = 1
Point-plane distance ∣ax1+by1+cz1+d∣a2+b2+c2\dfrac{\left\vert ax_1 + by_1 + cz_1 + d\right\vert}{\sqrt{a^2 + b^2 + c^2}}
Parallel planes ∣d1−d2∣a2+b2+c2\dfrac{\left\vert d_1 - d_2\right\vert}{\sqrt{a^2+b^2+c^2}} (normals matched first)
Plane-plane angle cos⁡θ\cos\theta of the normals
Line-plane angle sin⁡θ=∣b⃗⋅n⃗∣∣b⃗∣∣n⃗∣\sin\theta = \dfrac{\left\vert\vec{b}\cdot\vec{n}\right\vert}{\vert\vec{b}\vert\vert\vec{n}\vert}
Foot/image in plane walk P+tn⃗P + t\vec{n}, solve for tt; image at 2t2t
Family of planes P1+λP2=0P_1 + \lambda P_2 = 0 through the intersection line

The mistake checklist — run it before submitting

  1. Standardised the equation? Numerators x−x1x - x_1; no 3−x3 - x, no 2z2z, coefficients 11.
  2. Ratios vs cosines: divided by a2+b2+c2\sqrt{a^2+b^2+c^2} exactly when cosines were asked?
  3. All three λ\lambda's agree when testing a point on a line — two agreeing is not enough.
  4. Modulus in every distance and acute-angle formula.
  5. Sine for line-plane; cosine for line-line and plane-plane.
  6. Parallel-plane distance only after matching normals (scale one equation first).
  7. Foot before image: the image is 2Q−P2Q - P, never Q−PQ - P.
  8. Zero denominators in symmetric form are statements ("that coordinate is constant"), not divisions.

Where each skill was built

Direction cosines and ratios — Section 1. Line equations — Section 2. Angles — Section 3. Shortest distance — Section 4. Graded worked examples — Section 5. Board practice — Section 6. Planes and JEE tools — Section 7. Full JEE drill — Section 8.