Vectors & 3D Geometry · 100% marks notes

Method 01 · Components first

Stops abstract symbol chaos — every later product inherits clean components.

Do this: Write every vector as ⟨x,y,z⟩ or xi+yj+zk before algebra.

Method 02 · Displacement from points

Kills the yearly a+b joining-vector trap before it starts.

Do this: Always form AB = b−a when the stem gives coordinates.

Method 03 · Section formula lock

m,n order errors disappear when you write the ratio first.

Do this: Internal m:n → (n a + m b)/(m+n) — write m,n on rough sheet.

Method 04 · Equate components for combos

Linear combination stops being guesswork.

Do this: To write a in terms of b,c, solve the 2–3 component equations.

Magnitude / unit / displacement

Looks like: Unit vector along AB; |a|; vector joining points

Fast model: Form AB=b−a → |AB| → â=AB/|AB|.

Section / midpoint

Looks like: Point dividing a join in m:n

Fast model: Internal: (na+mb)/(m+n); write m,n first.

Linear combination

Looks like: Express a vector in terms of two others

Fast model: Equate i,j,k components → solve for coefficients.

Collinear / coplanar points via vectors

Looks like: Show points collinear; find λ so A,B,C collinear

Fast model: AB = k AC (or area |AB×AC|=0) for collinearity in plane/space.

Method 01 · Meaning before matrix

Choosing · vs × from the ask is the whole speed story.

Do this: Say angle? area? normal? — then pick · or ×.

Method 02 · Zero tests

⊥/∥ become one-line MC kills — use them before magnitudes.

Do this: Use a·b=0 / a×b=0 as first-line MC kills.

Method 03 · Determinant for cross

Cuts the classic j-component sign error.

Do this: Write the 3×3 ijk determinant every time — do not expand from memory mid-paper.

Method 04 · Area halves for triangles

Forgetting 1/2 is a free mark — this locks it.

Do this: Triangle area is half the parallelogram — write (1/2) explicitly.

Angle / perpendicular / parallel

Looks like: Find θ; show ⊥; show ∥

Fast model: · for angle/⊥; ×=0 (or proportional) for ∥.

Area parallelogram / triangle

Looks like: Area spanned by two vectors or three points

Fast model: |a×b|; triangle (1/2)|a×b|.

Projection

Looks like: Component / projection of a on b

Fast model: Scalar (a·b)/|b| or vector ((a·b)/|b|²)b.

Vector identity MC (· and × mix)

Looks like: (a×b)·(c×d); |a×b|²; Lagrange identity options

Fast model: Expand with ·/× identities; kill options with a tiny numeric triple.

Method 01 · STP as determinant

Coplanarity and volume become a clean 3×3 — less sign chaos.

Do this: Always compute [a b c] as a 3×3 det when components are given.

Method 02 · BAC–CAB for nested cross

Nested cross expansions stop exploding into component soup.

Do this: Write a×(b×c)=(a·c)b−(a·b)c immediately.

Method 03 · Volume = |STP|

Signed det is not the geometric volume — absolute value is.

Do this: Take absolute value after the determinant.

Method 04 · Points → edge vectors

Four-point coplanar checks stop using raw position vectors wrongly.

Do this: For four points, form AB,AC,AD from A, then STP.

Coplanar / volume / STP

Looks like: Three vectors coplanar? Parallelepiped volume?

Fast model: det; coplanar iff 0; volume=|det|.

Nested cross / BAC–CAB

Looks like: a×(b×c); simplify identity MC

Fast model: a×(b×c)=(a·c)b−(a·b)c.

Four points coplanar

Looks like: Points A,B,C,D coplanar?

Fast model: [AB AC AD]=0 using edge vectors from A.

Reciprocal / cyclic STP identities

Looks like: [a b c]=[b c a]; vanishing if two equal; linear dependence

Fast model: STP is alternating; two equal args ⇒ 0; treat each MC claim as T/F.

Method 01 · DR first

Every line formula is built from b — write it before the sheet of identities.

Do this: Always write b (direction) before angle or SD formulae.

Method 02 · Skew vs intersecting gate

Prevents the classic wrong use of skew SD on intersecting lines.

Do this: Compute (a₂−a₁)·(b₁×b₂) before calling lines skew.

Method 03 · Label a,b pairs

Wrong a₁/a₂ plugs die when labels are written first.

Do this: Write a₁,b₁,a₂,b₂ explicitly on rough sheet.

Method 04 · DC vs DR gate

Stops applying l²+m²+n²=1 to raw direction ratios.

Do this: If identity needs l²+m²+n²=1, convert DRs to DCs first.

Line forms & angle

Looks like: Equation of line; angle between two lines

Fast model: Extract b; cosθ=|b₁·b₂|/(|b₁||b₂|).

Skew shortest distance

Looks like: SD between two lines; show skew

Fast model: Gate STP≠0; then SD formula.

Coplanar / intersecting lines

Looks like: Are two lines coplanar? Find intersection

Fast model: (a₂−a₁)·(b₁×b₂)=0 ⇒ coplanar; then solve λ,μ.

Line of intersection of two planes

Looks like: Find line common to two planes; direction of intersection

Fast model: Direction b = n₁ × n₂; find one point by solving the system with a free variable.

Method 01 · Normal first

Every plane tool is a normal tool — n before distance/angle.

Do this: Write n before any plane distance/angle formula.

Method 02 · Plane from three points

Cross product normal beats a giant determinant mid-paper.

Do this: Two edges → n=v₁×v₂ → (r−a)·n=0.

Method 03 · Foot / image via λ

Reflection/foot asks become a bankable λ template.

Do this: Line through point parallel to n into plane; midpoint property for reflection.

Method 04 · Line–plane angle uses sin

Stops copying the planes cos formula onto line–plane asks.

Do this: Write sinφ=|b·n|/(|b||n|) — do not copy the planes cos formula.

Plane forms & three points

Looks like: Equation of plane; plane through three points

Fast model: n from cross of edges; (r−a)·n=0.

Distance / angles

Looks like: Point–plane distance; plane–plane or line–plane angle

Fast model: Distance formula; planes use cos of normals; line–plane uses sinφ.

Foot / image / intersection

Looks like: Foot of perpendicular; reflection; line meets plane

Fast model: P+μn into plane for foot; image=2F−P; line sub for intersection.

Family of planes / parallel plane

Looks like: Plane parallel to given plane through a point; π₁+λπ₂ family

Fast model: Parallel ⇒ same n; through point ⇒ plug into ax+by+cz+k=0. Family: π₁+λπ₂=0 for planes through intersection line.