Japan 高校2年 Mathematics C (vectors)
Chapters: 2
1. Vectors
Plane vectors and operations · Dot product · Vectors in figures · Space coordinates and vectors
- Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.
- Introduction to Three-dimensional Geometry (Class 11) – In space we use three mutually perpendicular axes x, y, z through the origin O. Each pair makes a coordinate plane: XY (z = 0), YZ (x = 0) and ZX (y = 0). The three planes divide space into eight octants, named by the signs of x, y, z. A point P(x, y, z) is reached by moving x along the x-axis, y parallel to the y-axis and z parallel to the z-axis; x, y, z are its distances from the YZ, ZX and XY planes. Points on the x-axis are (x, 0, 0); points on the XY-plane are (x, y, 0). The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is PQ = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²).
2. Mathematical representation (part)
Representing data with matrices and diagrams
- Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.