Class 11th mathematics(Introduction to three dimensional geometry)

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A line passing through the origin making angles p, q and r with x, y, z-axes then, the cosine of these angles, namely, cos p, cos q, and cos r are known as direction cosines of the line L. Any 3 numbers proportional to the direction cosines are known as the direction ratios of that line. If x, y, z are direction cosines and p, q, r are direction ratios of a line, then a = λl, b = λm, and c = λn, [where λ belongs to R].
If a, b, c are direction cosines of a line, then a2+b2+c2=1. Direction cosines of a line joining two points A(m1, n1, o1) and B(m2, n2, o2) are m2−m1AB=n2−n1AB=o2−o1AB
where PQ = (m2−m1)2+(n2−n1)2+(o2−o1)2−−−−−−−−−−−−−−−−−−−−√
Direction ratios of a line are the numbers which are proportional to the
Direction cosines of a line. If p, q, r are the direction cosines and m, n, o are the direction ratios of a line then,
p=mm2+n2+o2√q=nm2+n2+o2√r=om2+n2+o2√
If a1, b1, c1 and a2, b2, c2 are the direction cosines of two lines; and p is the acute angle between them; then cosp = |a1a2 + b1b2+ c1c2 |. The cartesian equation of a plane passing through the intersection of planes P1 x + Q1 y + R1 z + S1 = 0 and P2 x + Q2 y + R2 z + S2 = 0 is (P1 x + Q1 y + R1 z + S1) + l(P2 x + Q2 y + R2 z + S2) = 0. The distance from a point (x, y, z) to the plane Px1 + Qy1 + Rz1 + S = 0 is
|Px+Qy+Rz+S|P2+Q2+R2√

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