Line and Plane Questions with Solutions

This topic extends coordinate geometry into three dimensions. It covers direction cosines and direction ratios of a line, the equation of a line in vector and Cartesian form, the angle between two lines, the shortest distance between two skew lines, and the condition for two lines to intersect. It also covers the equation of a plane in its different forms (normal form, through a point perpendicular to a vector, through three points, and intercept form), the angle between two planes and between a line and a plane, the distance of a point from a plane, and the distance between parallel planes. Remember that the direction cosines l, m and n of any line satisfy l^2 + m^2 + n^2 = 1, and that the distance of a point (x1, y1, z1) from the plane ax + by + cz + d = 0 is |ax1 + by1 + cz1 + d| / sqrt(a^2 + b^2 + c^2). Practise the line and plane questions below, each with a step-by-step solution.