Scalar and vector products

The essentials

The scalar (dot) product of a and b is a · b = |a||b| cos θ, where θ is the angle between them (0 ≤ θ ≤ π). It is a number. In components, a · b = a₁b₁ + a₂b₂ + a₃b₃. Non-zero vectors are perpendicular exactly when a · b = 0. The projection of a on b is (a · b)/|b|.

The vector (cross) product a × b is a vector of magnitude |a||b| sin θ, perpendicular to both a and b, with direction given by the right-hand rule. In components it is the determinant with first row i, j, k, second row a₁, a₂, a₃ and third row b₁, b₂, b₃. Non-zero vectors are parallel exactly when a × b = 0.

Key points

• i · i = j · j = k · k = 1; i · j = j · k = k · i = 0.
• i × j = k, j × k = i, k × i = j; reversing the order changes the sign, for example j × i = −k.
• a · b = b · a, but a × b = −(b × a).
• Angle between vectors: cos θ = (a · b)/(|a||b|).
• Area of a parallelogram with adjacent sides a and b = |a × b|; area of a triangle with these sides = ½|a × b|.

Exam tips

• Use dot products for angles, perpendicularity and projections; use cross products for areas, perpendicular directions and parallelism.
• Expand the cross-product determinant carefully, remembering the minus sign on the j component.

Common mistakes

• Treating the dot product as a vector or the cross product as a number.
• Forgetting the ½ for the area of a triangle.