Vectors and their products
Vectors in NEB Class 12 include vector algebra, the scalar (dot) product, the vector (cross) product and the scalar triple product, with applications to angles, projections, areas and volumes. The exam tests both component calculation and geometric interpretation.
Core rules
- If vector a = a₁i + a₂j + a₃k and vector b = b₁i + b₂j + b₃k, then a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos θ.
- a ⟂ b if and only if a·b = 0. The projection of a on b is (a·b)/|b|.
- a × b is perpendicular to both; |a × b| = |a||b| sin θ. Area of a parallelogram is |a × b|; area of a triangle is half this.
- The scalar triple product [a b c] = a·(b × c) gives the signed volume of a parallelepiped; the vectors are coplanar exactly when it is zero.
- In components, a × b is the determinant with rows i, j, k; a₁, a₂, a₃; b₁, b₂, b₃.
Worked example
Let a = 2i - j + k and b = i + j + 2k. Then a × b = i[(-1)(2) - (1)(1)] - j[(2)(2) - (1)(1)] + k[(2)(1) - (-1)(1)] = -3i - 3j + 3k. Its magnitude is √(9 + 9 + 9) = 3√3, so the triangle formed by a and b has area 3√3/2.
Traps
- Using cos θ for a cross product or sin θ for a dot product.
- Reversing the order in a cross product, which changes the sign.
- Forgetting the factor 1/2 for triangle area.
- Treating a zero scalar triple product as a zero cross product; it means coplanarity, not perpendicularity.
- Mixing up the scalar projection with the projected vector.