{"id":19081,"date":"2022-08-06T15:42:34","date_gmt":"2022-08-06T10:12:34","guid":{"rendered":"https:\/\/apboardsolutions.com\/?p=19081"},"modified":"2023-03-30T15:06:04","modified_gmt":"2023-03-30T09:36:04","slug":"inter-1st-year-maths-1b-direction-cosines-and-direction-ratios-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.com\/inter-1st-year-maths-1b-direction-cosines-and-direction-ratios-formulas\/","title":{"rendered":"Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas"},"content":{"rendered":"

Use these Inter 1st Year Maths 1B Formulas<\/a> PDF Chapter 6 Direction Cosines and Direction Ratios to solve questions creatively.<\/p>\n

Intermediate 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas<\/h2>\n

Formulae and Synopsis :<\/strong>
\nIf a line PQ makes angles \u03b1, \u03b2, \u03b3 with the co – ordinate axes, then cos \u03b1, cos \u03b2, cos \u03b3 are called
\ndirection cosines of the line PQ. We take
\nl = cos \u03b1, m = cos \u03b2 and n = cos \u03b3
\nRelation between direction cosines is l2<\/sup> + m2<\/sup> + n2<\/sup> = 1<\/p>\n

Direction Ratios :<\/strong>
\nAn ordered triple of numbers proportional to the direction cosines of a line are defined as ‘Direction ratios’ of that fine.<\/p>\n

\u2192 If \\(\\frac{1}{a}=\\frac{m}{b}=\\frac{n}{c}\\) then a, b, c are called direction ratios of the line.<\/p>\n

\u2192 If a, b, c are the direction ratios of a line, then its direction cosines are \\(\\left(\\frac{a}{\\sqrt{a^{2}+b^{2}+c^{2}}}, \\frac{b}{\\sqrt{a^{2}+b^{2}+c^{2}}}, \\frac{c}{\\sqrt{a^{2}+b^{2}+c^{2}}}\\right)\\)<\/p>\n

\u2192 Direction ratios of the line joining P(x1<\/sub>, y1<\/sub>, z1<\/sub>) and Q (x2<\/sub>, y2<\/sub>, z2<\/sub>) are (x2<\/sub> – x1<\/sub>, y2<\/sub> – y1<\/sub>, z2<\/sub> – z1<\/sub>) or (x1<\/sub> – x2<\/sub>, y1<\/sub> – y2<\/sub>, z1<\/sub> – z2<\/sub>)<\/p>\n

\u2192 D.cs of the above line \\(\\frac{x_{2}-x_{1}}{P Q}, \\frac{y_{2}-y_{1}}{P Q}, \\frac{z_{2}-z_{1}}{P Q}\\) or \\(\\frac{x_{1}-x_{2}}{P Q}, \\frac{y_{1}-y_{2}}{P Q}, \\frac{z_{1}-z_{2}}{P Q}\\)<\/p>\n

\u2192 Angle between the lines whose D.cs are (l1<\/sub> m1<\/sub>, n1<\/sub>) and (l2<\/sub>, m2<\/sub>, n2<\/sub>) is given by cos \u03b8 = l1<\/sub>l2<\/sub> + m1<\/sub>m2<\/sub> + n1<\/sub>n2<\/sub><\/p>\n

\u2192 If these lines are perpendicular, then l1<\/sub>l2<\/sub> + m1<\/sub>m2<\/sub> + n1<\/sub>n2<\/sub> = 0<\/p>\n

\"Inter<\/p>\n

Angle Between Two Lines:<\/strong>
\nThe angle between two skew lines is the angle between two lines drawn parallel to them through any point in space.
\n\"Inter<\/p>\n

Direction Cosines:<\/strong>
\nIf \u03b1, \u03b2, \u03b3 are the angles made by a directed line segment with the positive directions of the coordinate axes respectively, then cos \u03b1, cos \u03b2, cos \u03b3 are called the direction cosines of the given line and they are denoted by l, m, n respectively Thus l = cos \u03b1, m = cos \u03b2, n = cos \u03b3
\n\"Inter<\/p>\n

The direction cosines of \\(\\overline{o p}\\) are l = cos \u03b1, m = cos \u03b2, n = cos \u03b3.
\nIf l, m, n are the d.c\u2019s of a line L is one direction then the d.c\u2019s of the same line in the opposite direction are -l, -m, -n.<\/p>\n

Note :<\/p>\n