Vectors in 3D, lines and planes · Year 13

In Year 13 you learn: vectors in 3D.

Vectors in three dimensions describe positions, directions and forces in space. The scalar product measures angles. The vector product finds perpendicular directions and areas. Vector equations describe lines and planes: the language of 3D graphics, engineering and physics.

Key facts

• |(a, b, c)| = √(a^2 + b^2 + c^2). The vector from A to B is b − a.
• Scalar product: a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ. a and b are perpendicular exactly when a · b = 0.
• Vector product: a × b is perpendicular to both, and |a × b| = |a||b| sin θ. The area of triangle ABC is 1/2 |AB × AC|.