HCA Mathematical Expositions: Determinants, cross products and the scalar triple product

© Gerard Sayson, 2025 1 Determinants, cross products and the scalar triple product Determinants, cross products and the scalar triple product Exploring the cross product in greater detail Gerard Sayson <geruls@broskiclan.org > January 27, 2025 Preface *looks at MF27* What is this crappy formula for the cross product? GERARD SAYSON Ask most H2 Mathematics students to give you the formula for a cross product. You’ll be seeing one of two types of these students: • The student can memorize and regurgitate the required answer. • They either scavenge through their formula booklet or are unable to come up with the answer. According to MF27, the official formula booklet, we have the following: [ 𝑎1 𝑎2 𝑎3 ] × [ 𝑏1 𝑏2 𝑐3 ] = [ 𝑎2𝑏3 − 𝑎3𝑏2 𝑎3𝑏1 − 𝑎1𝑏3 𝑎1𝑏2 − 𝑎2𝑏1 ] Would you look at that?! It’s hard to memorize , hard to derive and just outright confusing! Instead of that mess, what if I told you that there’s a way to memorize it much more efficiently? Don’t believe me? Have a look below: [ 𝑎1 𝑎2 𝑎3 ] × [ 𝑏1 𝑏2 𝑐3 ] = | 𝐢 𝐣 𝐤 𝑎1 𝑎2 𝑎3 𝑏1 𝑏2 𝑏3 | Now, you may be wondering: what is that? What are those vertical lines? No, they’re not the absolute value; you can’t take that for a matrix! Instead, we have what’s called the determinant. With the determinant, we can calculate something explicitly excluded from the syllabus: the scalar triple product , 𝐚 ⋅ (𝐛 × 𝐜). In fact, with the determinant, we can give it a geometric meaning and enhance our knowledge of vectors! This document was written in Microsoft Word instead of the usual LaTeX, because making diagrams in LaTeX is a pain! Also, check out https://lib.gsn.bz for more stuff.

© Gerard Sayson, 2025 2 Determinants, cross products and the scalar triple product What is a determinant? To put it simply, a determinant of a square matrix, denoted by det 𝐀, is a function of the square matrix which has some special properties. If 𝐀 is two-dimensional, we have det 𝐀 = |𝑎 𝑏 𝑐 𝑑| = 𝑎𝑑 − 𝑏𝑐 where 𝐀 = [𝑎 𝑏 𝑐 𝑑]. But what is the geometric significance? Suppose we have a vector 𝐯 = [𝑥 𝑦]. We define a linear transformation 𝑇 as a function that, for any vectors 𝐮 and 𝐯, the properties 𝑇(𝐮 + 𝐯) = 𝑇(𝐮) + 𝑇(𝐯) 𝑇(𝑐𝐮) = 𝑐𝑇(𝐮) are true. For example, the transformation 𝑥′ = 𝑥 cos(𝜃) − 𝑦 sin(𝜃) 𝑦′ = 𝑥 sin(𝜃) + 𝑦 cos(𝜃) is linear. The above transformation corresponds to rotating a point about the origin by 𝜃: Note that 𝑟 = √𝑥2 + 𝑦2 and tan(𝛼) = 𝑦 𝑥. This allows us to consider the angle and the length of the line from the origin. See that 𝑥′ = 𝑟 cos(𝛼 + 𝜃) = 𝑟 cos(𝛼) cos(𝜃) − 𝑟 sin(𝛼) sin(𝜃) = 𝑥 cos(𝜃) − 𝑦 sin(𝜃) 𝑦′ = 𝑟 sin(𝛼 + 𝜃) = 𝑟 sin(𝛼) cos(𝜃) + 𝑟 cos(𝛼) sin(𝜃) = 𝑥 sin(𝜃) + 𝑦 cos(𝜃). Is this a linear transformation? Yes, it is. Exercise 1. Verify that this is a linear transformation. Note that we can represent any linear transformation as a matrix: 𝑅𝜃 = [cos(𝜃) − sin(𝜃) sin(𝜃) cos(𝜃) ] ⟺ 𝑅𝜃 [𝑥 𝑦] = [𝑥 cos(𝜃) − 𝑦 sin(𝜃) 𝑥 sin(𝜃) + 𝑦 cos(𝜃)] This key result holds due