Vector Algebra
Vector Algebra covers directed quantities, addition and scalar multiplication, the scalar and vector products, the scalar triple product, and geometric applications such as area and coplanarity. The exam usually asks for a computation with a small amount of interpretation.
Products and their uses
The dot product a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃ is a scalar, and a·b = 0 exactly when the vectors are perpendicular. The cross product a×b = |a||b| sin θ n̂ is a vector perpendicular to both, with direction given by the right-hand rule; a×b = 0 when the vectors are parallel. In components, a×b is the determinant with i, j, k in the first row. Then |a×b| is the area of the parallelogram on the two vectors, and half of it is the area of the triangle. The scalar triple product [a b c] = a·(b×c) gives the volume of the parallelepiped, and the vectors are coplanar exactly when it vanishes.
Worked example
Take a = 2i + 3j − k and b = i − j + 2k. Then a·b = 2 − 3 − 2 = −3. Expanding the determinant, a×b = i(6 − 1) − j(4 + 1) + k(−2 − 3) = 5i − 5j − 5k, whose magnitude is 5√3. The triangle with a and b as adjacent sides therefore has area (5√3)/2.
Traps
- Forgetting the minus sign on the j-component of a cross product.
- Dropping the factor ½ in the triangle area.
- Confusing perpendicular (dot product zero) with parallel (cross product zero).
- Treating the scalar triple product as a vector.