Lines in space: direction vectors, three forms of equation and the angle between lines

In a plane a line is fixed by its slope and one point. In space, one number is not enough to describe a tilt, so we use a whole vector. A line in space is fixed by one point A on it and one direction. A direction vector d is any non-zero vector parallel to the line: if A and B lie on the line, d = AB. Every non-zero multiple of d is also a direction vector, so the components a, b, c of d, called the direction ratios (in Arabic نسب الاتجاه), matter only through their ratios. The direction cosines of Week 9 are the components of the unit vector d/‖d‖; so ℓ² + m² + n² = 1 and the direction ratios are proportional to ℓ, m, n. To go from ratios to cosines, divide by ‖d‖. The axes have direction vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1). A zero in d tells you something: if the y-component is 0, the y-coordinate never changes along the line, so the line is parallel to the xz-plane. Now the vector equation. Take any point of the line with position vector r = (x, y, z). Starting at A and moving along the line by a multiple of d, we get r = A + t d, where t is any real number, called the parameter. Each value of t gives one point of the line: t = 0 gives A, t = 1 gives A + d, and a negative t gives points on the other side of A. The same line has many vector equations, because we may choose a different point A or a different multiple of d.