⁦Vectors in space⁩: ⁦length, distance and direction cosines⁩

⁦This studio follows Afghan Year 11 chapter 7 (vectors in space), Iran⁩'⁦s Year 12 geometry chapter 3 and A level vectors⁩. ⁦In 3D coordinates every point has three coordinates and a⁩ شعاع ⁦is written a = xi + yj + zk⁩. ⁦For example (٢، ٣، ٦) is ٢i + ٣j + ٦k⁩.

⁦The length of a vector is the diagonal of a cuboid whose edges are its components⁩: |a| = √(x^٢ + y^٢ + z^٢). ⁦The distance between points A and B is the length of AB = OB − OA, and each coordinate of the midpoint is the mean of the matching coordinates⁩.

⁦If α, β and γ are the angles the vector makes with the x, y and z axes, each cosine is that axis⁩'⁦s component divided by the length⁩. ⁦The squares of the three cosines add up to 1⁩. ⁦Common mistakes⁩: ⁦adding the components instead of their squares, forgetting the square root, and subtracting coordinates in the wrong order⁩.