( vi) The projection of the line segment joining points P( x1, y1, z1) and Q( x2, y2, z2) to the line having direction cosines 1, m, n is |( x2 – x1) l +( y2 – y1) m +( z2 – z1) n |.
( vii) The direction ratio of the line passing through points P( x1, y1, z1) and Q( x2, y2, z2) are proportional to x2 – x1, y2 – y1 – z2 – z1 Then, direction cosines of PQ are
x2 – x1 / | PQ |, y2 – y1 / | PQ |, z2 – z1 / | PQ |
Area of Triangle
If the vertices of a triangle be A( x1, y1, z1) and B( x2, y2, z2) and C( x3, y3, z3), then
Angle Between Two Intersecting Lines
If l( x1, m1, n1) and l( x2, m2, n2) be the direction cosines of two given lines, then the angle θ between them is given by
cos θ = l112 + m1m2 + n1n2( i) The angle between any two diagonals of a cube is cos-1( 1 / 3).
( ii) The angle between a diagonal of a cube and the diagonal of a face( of the cube is cos-1( √2 / 3)
Straight Line in Space
The two equations of the line ax + by + cz + d = 0 and a’ x + b’ y + c’ z + d’ = 0 together represents a straight line.
1. Equation of a straight line passing through a fixed point A( x1, y1, z1) and having direction ratios a, b, c is given by