Geometric Bites

Linear Transformations in ℝ³ and 3×3 Matrices Matrices give us a compact way to describe linear transformations in three-dimensional space. A linear transformation is a mapping T : ℝ³ → ℝ³ that sends a point with position vector (x, y, z) to another point, according to a rule with two key properties. What Makes a Transformation Linear? A transformation T : ℝ³ → ℝ³ is called linear if, for all real numbers λ and all vectors (x, y, z) in ℝ³, T(λx, λy, λz) = λ T(x, y, z), and for all vectors (x₁, y₁, z₁) and (x₂, y₂, z₂) in ℝ³, T(x₁ + x₂, y₁ + y₂, z₁ + z₂) = T(x₁, y₁, z₁) + T(x₂, y₂, z₂). The point that (x, y, z) is sent to is called the image of (x, y, z) under T. The Standard Basis Vectors To find the matrix that represents a particular transformation, it is enough to know what happens to three special vectors, called the standard basis for ℝ³: î = (1, 0, 0) ĵ = (0, 1, 0) k̂ = (0, 0, 1) Once we know the images of î, ĵ and k̂, th...

Converting the Vector Equation of a Line into Cartesian Form A straight line in three-dimensional space can be expressed using vectors. One important vector form is (𝐑 − 𝐀) × 𝐁 = 0 This equation states that the displacement vector from a fixed point 𝐀 to a general point 𝐑 is parallel to the direction vector 𝐁. Two non-zero vectors have a zero cross product precisely when they are parallel. From this fact, the Cartesian (symmetric) equation of the line can be derived. 1. Substituting Coordinate Vectors The general point on the line is written as 𝐑 = (x, y, z) The fixed point is 𝐀 = (x₁, y₁, z₁) The direction vector is 𝐁 = (l, m, n) Substituting these into the vector equation yields: ((x, y, z) − (x₁, y₁, z₁)) × (l, m, n) = 0 which simplifies to: (x − x₁, y − y₁, z − z₁) × (l, m, n) = 0 2. Using the Condition for a Zero Cross Product If two non-zero vectors have a zero cross product, then one is a scalar multiple of the other. T...