Physics · Mathematical methods and mechanics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
A vector has size and direction. Two products matter. The dot product A·B = AB cos θ is a scalar, and it is zero for perpendicular vectors. The cross product A × B is a vector of size AB sin θ, perpendicular to both A and B (right-hand rule), and it is zero for parallel vectors.
For fields we use the operator ∇. The gradient of a scalar field points in the direction of steepest increase. Divergence tells how much a vector field spreads out from a point. Curl tells how much it circulates round a point.
Gauss's theorem links the flux out of a closed surface to the divergence inside. Stokes' theorem links the line integral round a closed loop to the curl through it.
Mixing up the two zero tests. A·B = 0 means the vectors are perpendicular; A × B = 0 means they are parallel. The order also matters in a cross product: j × i = −k.
Answer: θ = 45° and the area is 5 square units.
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