. . Angle between two straight lines (a) Vector form The acute angle between two given straight lines r sb and r td is same as that of the angle between b and d . So, | | b d or | | b d − Remark (i) The two given lines r sb and r td are parallel Û θ = Û cos θ = Û | | | | | b d (ii) The two given lines r sb and r td are parallel if, and only if b , for some scalar λ .
(iii) The two given lines r sb and r td are perpendicular if, and only if b d (b) Cartesian form If two lines are given in Cartesian form as then the acute angle θ between the two given lines is given by θ = b d b d b d Remark (i) The two given lines with direction ratios b b b and d d d are parallel if, and only if (ii) The two given lines with direction ratios b b b and d d d are perpendicular if and only if b d b d b d (iii) If the direction cosines of two given straight lines are l m n and l m n , then the angle between the two given straight lines is l l m m n n θ = Vector - - Example . Find the acute angle between the lines ) ( and the straight line passing through the points ( , , ) and ( , , ) . We know that the line ) ( is parallel to the vector Direction ratios of the straight line joining the two given points ( , , ) and ( , , ) are , , and hence this line is parallel to the vector Therefore, the acute angle between the given two straight lines is θ = | | b d − , where Therefore, θ = | ( ) ( ) | | | |