Transformations and chapter review

Main formulas of the chapter: AB = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²); the midpoint is the half-sums of the coordinates; the vector AB = (x₂ − x₁, y₂ − y₁, z₂ − z₁); |a| = √(a₁² + a₂² + a₃²); a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos φ; a ⊥ b ⇔ a · b = 0; a ∥ b ⇔ proportional coordinates; sphere: (x − a)² + (y − b)² + (z − c)² = R². Three points lie on a line if AB and AC are collinear; a quadrilateral is a parallelogram if AB = DC or if the diagonals have the same midpoint. In test questions first decide which formula is needed and then check the answer options by computation, not by guessing; for perpendicularity set the dot product equal to zero. For symmetries recall which coordinate signs change: for a plane — one, for an axis — two, for a point — three.