How to Calculate the Cross Product in 3D Vectors Definition, Types, Examples & Uses

How to Calculate the Cross Product in 3D Vectors

Subject: Mathematics

Book: Maths Mastery

For 3D vectors u20d7A=[a₁,a₂,a₃] and u20d7B=[b₁,b₂,b₃], their cross product u20d7A×u20d7B is [a₂b₃–a₃b₂, a₃b₁–a₁b₃, a₁b₂–a₂b₁]. This new vector is perpendicular to both u20d7A and u20d7B, with magnitude = |A||B|sin(θ). Cross products are fundamental in calculating torque, normal vectors for surfaces, or rotation axes in physics. For instance, with u20d7A=[1,2,3] and u20d7B=[4,5,6], u20d7A×u20d7B=[(2×6–3×5), (3×4–1×6), (1×5–2×4)]=[12–15, 12–6, 5–8]=[–3,6,–3]. Proficiency ensures advanced 3D geometry or mechanical tasks are handled with rigor and clarity.