( i) If 1, m, n are direction cosines of a vector r, then( a) r = | r |( li + mj + nk) ⇒ r = li + mj + nk( b) l 2 + m 2 + n 2 = 1( c) Projections of r on the coordinate axes are
( d) | r | = l | r |, m | r |, n | r | / √sum of the squares of projections of r on the coordinate axes
( ii) If P( x1, y1, z1) and Q( x2, y2, z2) are two points, such that the direction cosines of PQ are l, m, n. Then,
x2 – x1 = l | PQ |, y2 – y1 = m | PQ |, z2 – z1 = n | PQ | These are projections of PQ on X, Y and Z axes, respectively.
( iii) If 1, m, n are direction cosines of a vector r and a b, c are three numbers, such that l / a = m / b = n / c.
Then, we say that the direction ratio of r are proportional to a, b, c. Also, we have l = a / √a 2 + b 2 + c 2, m = b / √a 2 + b 2 + c 2, n = c / √a 2 + b 2 + c 2
( iv) If θ is the angle between two lines having direction cosines l1, m1, n1 and 12, m2, n2, then
cos θ = l112 + m1m2 + n1n2( a) Lines are parallel, if l1 / 12 = m1 / m2 = n1 / n2( b) Lines are perpendicular, if l112 + m1m2 + n1n2
( v) If θ is the angle between two lines whose direction ratios are proportional to a1, b1, c1 and a2, b2, c2 respectively, then the angle θ between them is given by
cos θ = a1a2 + b1b2 + c1c2 / √a 2 1 + b 2 1 + c 2 1 √a 2 2 + b 2 2 + c 2 2 Lines are parallel, if a1 / a2 = b1 / b2 = c1 / c2 Lines are perpendicular, if a1a2 + b1b2 + c1c2 = 0.