Vectors on a prism and a cuboid: equal, sums, collinear
Tell equal, codirectional and opposite vectors apart on a prism, add vectors by the triangle rule, express one vector through others and find collinear vectors by coordinates.
- Ukraine: grade 10
- Russia: grade 9
Guide
From the Ukrainian Grade 10 textbook (algebra and calculus with geometry): §6 'Coordinates and vectors in space', items 39 to 41. A vector is a directed segment, written AB (A is the start, B the end) or with one small letter a. The modulus of AB is the length of the segment AB, written |AB|.
Two non-zero vectors are collinear if they lie on parallel lines or on one line. Collinear vectors are codirectional (AO ↑↑ A₁C₁) or opposite (OD ↑↓ D₁B₁). Two non-zero vectors are equal if their moduli are equal and they are codirectional.
The triangle rule: for any three points A, B and C, AB + BC = AC. To add vectors, replace them by equal ones so that the end of the first is the start of the next. The difference: OA − OB = BA.
The product of a non-zero vector a and a number k ≠ 0 is the vector b = ka with |b| = |k| · |a|; if k > 0 then b ↑↑ a, and if k < 0 then b ↑↓ a. Vectors a and b are collinear exactly when b = ka; then their coordinates are proportional.