12 min readConcept sheet1-mark MCQ; 2-mark and 3-mark questions on area and on a unit vector perpendicular to two vectors; also a JEE favourite

Vector (cross) product

a⃗ × b⃗ = |a⃗| |b⃗| sin θ n̂ is a vector perpendicular to both. Its magnitude is the area of the parallelogram with sides a⃗ and b⃗.

Definitiona⃗ × b⃗ = |a⃗| |b⃗| sin θ n̂, where n̂ is the unit vector perpendicular to a⃗ and b⃗ given by the right-hand rule

Determinant forma⃗ × b⃗ = (a₂b₃ − a₃b₂)î − (a₁b₃ − a₃b₁)ĵ + (a₁b₂ − a₂b₁)k̂

AreaParallelogram with adjacent sides a⃗, b⃗: |a⃗ × b⃗|. Triangle: (1/2)|a⃗ × b⃗|. Extra help (not in your textbook): a parallelogram with diagonals d⃗₁, d⃗₂ has area (1/2)|d⃗₁ × d⃗₂|

Parallel testFor non-zero a⃗, b⃗: a⃗ × b⃗ = 0⃗ exactly when a⃗ ∥ b⃗

Definition, right-hand rule and meaning

The vector (cross) product of a⃗ and b⃗ is a⃗ × b⃗ = |a⃗| |b⃗| sin θ n̂, where 0 ≤ θ ≤ π. Here n̂ is a unit vector perpendicular to both a⃗ and b⃗. Its sense comes from the right-hand rule: curl the fingers of your right hand from a⃗ towards b⃗, and the thumb points along n̂. The result is a vector. Its magnitude |a⃗| |b⃗| sin θ equals base × height of the parallelogram with sides a⃗ and b⃗, so |a⃗ × b⃗| is the area of that parallelogram. In components, expand the determinant with rows (î, ĵ, k̂), (a₁, a₂, a₃) and (b₁, b₂, b₃).

The î, ĵ, k̂ cycle

back to î

Going round the cycle gives a plus sign. Going backwards gives a minus sign, e.g. ĵ × î = −k̂. Also î × î = ĵ × ĵ = k̂ × k̂ = 0⃗.

Properties

Not commutative: b⃗ × a⃗ = −(a⃗ × b⃗).
Distributive: a⃗ × (b⃗ + c⃗) = a⃗ × b⃗ + a⃗ × c⃗.
λ(a⃗ × b⃗) = (λa⃗) × b⃗ = a⃗ × (λb⃗).
a⃗ × a⃗ = 0⃗, and a⃗ × b⃗ = 0⃗ when a⃗ and b⃗ are parallel.
Unit vector perpendicular to a⃗ and b⃗: ±(a⃗ × b⃗)/|a⃗ × b⃗|.
Extra help (not in your textbook): |a⃗ × b⃗|² + (a⃗ · b⃗)² = |a⃗|² |b⃗|², because sin²θ + cos²θ = 1.

Dot product vs cross product

a⃗ · b⃗A scalar: |a⃗| |b⃗| cos θ. Commutative. Zero when a⃗ ⊥ b⃗. Gives angles and projections.

a⃗ × b⃗A vector: |a⃗| |b⃗| sin θ n̂. Changes sign when the order is swapped. Zero when a⃗ ∥ b⃗. Gives areas and perpendicular directions.

Cosine goes with dot, sine goes with cross.

Example 1: a unit vector perpendicular to two vectors

Problem

Find a unit vector perpendicular to both a⃗ = 2î − ĵ + k̂ and b⃗ = 3î + 4ĵ − k̂.

Working

a⃗ × b⃗ = |î ĵ k̂; 2 −1 1; 3 4 −1|
î part: (−1)(−1) − (1)(4) = 1 − 4 = −3
ĵ part: −[(2)(−1) − (1)(3)] = −(−2 − 3) = 5
k̂ part: (2)(4) − (−1)(3) = 8 + 3 = 11
a⃗ × b⃗ = −3î + 5ĵ + 11k̂, and |a⃗ × b⃗| = √(9 + 25 + 121) = √155
Check: a⃗ · (a⃗ × b⃗) = −6 − 5 + 11 = 0 and b⃗ · (a⃗ × b⃗) = −9 + 20 − 11 = 0.

Answer

±(−3î + 5ĵ + 11k̂)/√155

Example 2: area of a triangle

Problem

Find the area of the triangle with vertices A(1, 2, 0), B(3, 1, 2) and C(2, 4, 1).

Working

AB⃗ = 2î − ĵ + 2k̂ and AC⃗ = î + 2ĵ + k̂
AB⃗ × AC⃗: î part (−1)(1) − (2)(2) = −5
ĵ part: −[(2)(1) − (2)(1)] = 0
k̂ part: (2)(2) − (−1)(1) = 5
AB⃗ × AC⃗ = −5î + 5k̂, so |AB⃗ × AC⃗| = √50 = 5√2
Area = (1/2)|AB⃗ × AC⃗| = 5√2/2

Answer

Area = (5√2)/2 square units.

In real life

Turning a spanner on a bicycle nut produces a torque τ⃗ = r⃗ × F⃗. Pushing at right angles to the handle, far from the nut, gives the biggest turning effect.

Exam angle

1 marks: “Find ĵ × (2î + k̂).” ĵ × 2î = −2k̂ and ĵ × k̂ = î. So the answer is î − 2k̂.

2 marks: “If |a⃗| = 2, |b⃗| = 5 and a⃗ · b⃗ = 8, find |a⃗ × b⃗|.” cos θ = (a⃗ · b⃗)/(|a⃗| |b⃗|) = 8/10 = 4/5. As 0 ≤ θ ≤ π, sin θ = 3/5. So |a⃗ × b⃗| = |a⃗| |b⃗| sin θ = 2 × 5 × 3/5 = 6.

3 marks: “Find the area of the parallelogram whose adjacent sides are a⃗ = 2î + ĵ − k̂ and b⃗ = î − 3ĵ + 2k̂. Also find a unit vector perpendicular to both.” a⃗ × b⃗ = −î − 5ĵ − 7k̂, with magnitude √75 = 5√3. Area = 5√3 square units. Unit vector perpendicular to both = ±(î + 5ĵ + 7k̂)/(5√3).

Remember it

Cross gives a vector at right angles; its size is the parallelogram's area.