( 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 .