Vectors
Vector algebra in the IPE papers covers addition and scalar multiplication, the scalar and vector products, the scalar triple product, coplanarity, vector equations of lines and planes, and geometric applications such as area of a triangle and volume of a parallelepiped.
Core rules and concepts
- a·b = |a||b| cos θ; a×b = |a||b| sin θ n̂, with n̂ perpendicular to both and following the right-hand rule.
- i·i = 1, i·j = 0, i×i = 0, i×j = k, j×k = i, k×i = j.
- a×b = −b×a; the magnitude |a×b| is the area of the parallelogram on a and b.
- Scalar triple product [a b c] = a·(b×c); its modulus is the volume of the parallelepiped, and [a b c] = 0 exactly when a, b, c are coplanar.
- Projection of a on b is (a·b)/|b|; the unit vector along a is a/|a|.
Worked example
Find λ so that a = i + j + k, b = 2i + 3j + k and c = i + λj + 3k are coplanar. Set the triple product to zero: |1 1 1; 2 3 1; 1 λ 3| = 1(9 − λ) − 1(6 − 1) + 1(2λ − 3). Simplify: 9 − λ − 5 + 2λ − 3 = 1 + λ. So 1 + λ = 0, giving λ = −1.
Traps that cost marks
- Reversing the order in a cross product and losing the sign.
- Reporting a vector where the question asks for a scalar (dot product and triple product are scalars).
- Using an unnormalised vector when a unit vector or a direction cosine is demanded.
- Treating a·b = 0 as perpendicularity only for non-zero vectors; check for zero vectors.