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            ANGLE BETWEEN TWO LINES | 3-DIMENSIONAL 
                 GEOMETRY | NCERT CLASS 12 MATHS 
         
         
        In accordance with NCERT Class 12 Maths, 3D geometry alludes to the mathematics of 
        shapes in three-dimensional space and comprises of 3 coordinates. These 3 coordinates are 
        x-coordinate, y-coordinate, and z-coordinate. In three-dimensional space, there is a 
        necessity of three parameters to locate the specific area of a point. Dimension, in like 
        manner speech, signifies the proportion of an item's size, for example, a box, generally 
        given as height, length, and breadth. In geometry, the thought of dimension is an 
        augmentation of the possibility that a line speaks to one-dimensional, a plane happens to 
        be two-dimensional, and space is three-dimensional.  
         
        The arrangement of a three-dimensional Cartesian coordinate system is referred to as the 
        origin just as a premise including three mutually perpendicular vectors with respect to 
        NCERT Class 12 Maths. These vectors appropriately clarify the three coordinate axes which 
        are: the x-, y-, and z-axis. Specialists additionally call them as abscissa, ordinate and 
        applicate pivot, separately. 
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            ANGLE BETWEEN TWO LINES 
             
            According to NCERT Class 12 Maths, Angle between two lines alludes to the angle between 
            two intersecting lines. This is because the angle between the two perpendicular lines is 90° 
            and that angle between two parallel lines will be 0°. Thus, we will presently take a gander 
            at how the angle between two lines is determined. 
             
             
            CARTESIAN FORM 
            Let ܮ  and ܮ be two lines passing through the origin and with direction ratios 
               ଵ     ଶ
            ܽ ,ܾ ,ܿ andܽ ,ܾ ,ܿ , respectively. Let P be a point on ܮ and Q be a point onܮ . Consider the 
             ଵ ଵ ଵ    ଶ ଶ  ଶ                           ଵ                ଶ
            directed lines OP and OQ as given in the following figure. Let θ be the acute angle between 
            OP and OQ. Now recall that the directed line segments OP and OQ are vectors with 
            components ܽ ,ܾ ,ܿ and ܽ ,ܾ ,ܿ , respectively. Therefore, the angle between two lines 
                       ଵ ଵ ଵ     ଶ ଶ ଶ
            formula is given by: 
             
            ANGLE BETWEEN TWO LINES FORMULA: 
             
                                          ࢇ ࢇ +࢈ ࢈ +ࢉ ࢉ
                                           ૚ ૛     ૚ ૛    ૚ ૛
                                     ተ                           ተ
                                 ࣂ=                                
                                     ተ                           ተ
                             ܋ܗܛ
                                         ૛    ૛   ૛   ૛    ૛    ૛
                                      ටࢇ +࢈ +ࢉ ටࢇ +࢈ +ࢉ
                                         ૚    ૚   ૚   ૛    ૛    ૛
             
             
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                            Here, equations of the two lines are of form: 
                             
                                                                                     ܠ−ܠ                      ܡ−ܡ                     ܢ−ܢ
                                                                                                   ૚ =                     ૚ =                     ૚ 
                                                                                          ܉                       ܊                        ܋
                                                                                             ૚                        ૚                       ૚
                                                                                     ܠ−ܠ                      ܡ−ܡ                     ܢ−ܢ
                                                                                                   ૛ =                     ૛ =                     ૛ 
                                                                                          ܉                       ܊                        ܋
                                                                                             ૛                        ૛                       ૛
                             
                            VECTOR FORM 
                                                                                                                 ሬሬሬ⃗                               ሬሬሬሬ⃗
                            Let the equations of two lines be ݎ⃗ = ܽሬሬሬሬ⃗ + ߣܾ  and ݎ⃗ = ܽሬሬሬሬ⃗ + ߣܾ  such that ߠ denotes angle 
                            between the two lines.                                                     ଵ           ଵ                      ଶ           ଶ
                             
                            Then, Angle between two lines formula will be: 
                             
                                                                                                              ࢈ ⋅࢈
                                                                                    ܋ܗܛࣂ = ฬ ૚ ૛ ฬ 
                                                                                                              ሬሬሬሬ⃗  ሬሬሬሬ⃗
                                                                                                            ห࢈ หห࢈ ห
                                                                                                                ૚       ૛
                             
                             
                            VECTOR AND CARTESIAN EQUATIONS OF A LINE 
                             
                                                                                                                        ሬሬሬ⃗                                                         ሬ⃗
                                   1.  Equation of a line through a given point                                            and parallel to a given vector ܾ  is given by 
                                                           ሬ⃗
                                          ݎ⃗ = ܽ⃗ + ߣܾ where ߣ denotes any parameter. 
                                          In Cartesian Form: 
                                                                                                                      (                )
                                          Let the coordinates of the given point A be  ݔ ,ݕ ,ݖ and the direction ratios of the line are 
                                                                                                                          ଵ     ଵ    ଵ
                                           a, b, c. Consider the coordinates of any point P be (x, y, z). Then the Cartesian Equation of a 
                                          line is   
                                           
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                                                                                   ܠିܠ૚ = ܡିܡ૚ = ܢିܢ૚ 
                                                                  ܉           ܊           ܋
                             
                        2.  Equation of a line passing through two given points: 
                                                                                                              
                                                      ሬ⃗
                            Therefore, ݎ⃗ = ܽ⃗ + ߣ(ܾ − ܽ⃗). 
                            In Cartesian Form, the Cartesian Equation will be: 
                                                      
                                                          ࢞ି࢞૚  = ࢟ି࢟૚ = ࢠିࢠ૚ 
                                                      ࢞૛ି࢞૚         ࢟૛ି࢟૚          ࢠ૛ିࢠ૚
                    
                    
                    
                    
                    
                   IMPORTANT FORMULAE OF 3-DIMENSIONAL GEOMETRY WITH RESPECT 
                                                        TO NCERT CLASS 12 MATHS 
                    
                    
                    
                        1.  If ܽ, ܾ,ܿ denote direction ratios of line then direction cosines are: 
                         
                                ܔ = ±               ܉            , ܕ = ±                ܊            , ܖ = ±               ܋            
                                          √ ૛         ૛       ૛               √ ૛         ૛       ૛              √ ૛         ૛       ૛
                                            ܉ +܊ +܋                             ܉ +܊ +܋                            ܉ +܊ +܋
                             
                             
                             
                             
                        2.  If ݈, ݉, ݊ denote direction ratios of the line then ݈ଶ + ݉ଶ + ݊ଶ = 1 
                             
                             
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