ACJC 2025 Vectors II Lecture Notes

1 6B THREE-DIMENSIONAL VECTOR GEOMETRY SYLLABUS • Vector and cartesian equations of lines and planes • Foot of the perpendicular and distance from a point to a line or to a plane • Angle between two lines, between a line and a plane, or between two planes • Relationships between - two lines (coplanar or skew) - a line and a plane - two planes

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Lines ................................................................................................. 3 1.1 Vector Equation of a Line ....................................................... 3 1.2 Parametric and Cartesian Equations of a Line ....................... 5 2 Relationships between Point and Line ............................................. 9 2.1 Foot of Perpendicular from a Point to a Line ......................... 9 2.2 Perpendicular Distance from a Point to a Line .................... 12 3 Relationships between Two Lines ................................................. 16 3.1 Parallel, Intersecting or Skew Lines .................................... 16 3.2 Angle between Two Lines ................................................... 23 4 Planes ............................................................................................. 25 4.1 Vector Equation of a Plane in Parametric Form ................... 25 4.2 Vector Equation of a Plane in Scalar Product Form ............ 27 4.3 Cartesian Equation of a Plane .............................................. 31 4.4 Special Planes ...................................................................... 32 5 Relationships between Point and Plane ......................................... 34 5.1 Point on a Plane .................................................................... 34 5.2 Foot of Perpendicular from a Point to a Plane ..................... 35 5.3 Perpendicular Distance from a Point to a Plane .................... 38 5.4 Distance of the Origin O from a Plane ................................. 39 5.5 Reflection of a Point in a Plane ............................................ 40 6 Relationships between Line and Plane .......................................... 41 6.1 Possible Configurations of a Line and a Plane .................... 41 6.2 Intersection of a Line and a Plane ........................................ 42 6.3 Angle between a Line and a Plane ....................................... 43 6.4 Reflection of a Line in a Plane ............................................. 45 7 Relationships between Two Planes ............................................... 46 7.1 Two Parallel Planes .............................................................. 46 7.2 Intersection of Two Planes ................................................... 48 7.3 Angle between Two Planes .................................................. 52 Annex A: Practice Questions on Lines and Planes ................................. 55

6B Vectors II 3 LECTURE 1 Lesson Outline • Vector equation of a line • Parametric and Cartesian equations of a line 1 LINES 1.1 Vector Equation of a Line Consider an origin O, a point A with position vector a, a vector b and a line l that passes through point A and is parallel to b. Let 1P , 2P and 3P be three points on the line. Suppose 1AP = b . Then 11OP OA AP= + = + ab . Suppose 5 2 2AP = b . Then 5 22 2OP OA AP= + = + ab . Suppose 3 3AP =− b . Then 33 3OP OA AP= + = − ab O A a b O A a b O A a b

ACJC 2025/26 H2 Mathematics (9758) 4 Combining the 3 cases, we have In general, i f P is any point on the line l such that AP = b , then the position vector r of P is given by OP OA AP = = + = +r a b . Conversely, the vector =+r a b , where   , is the position vector of a point P on the line passing through A and parallel to b such that .AP = b The vector b is called the direction vector of the line. Hence, the vector equation of a line l passing through point A with position vector a and parallel to a vector b is =+r a b , where  is a parameter,   . Note • Expression of the equation of a line is not unique. • If the equation of the line can be written as =rb , where  is a parameter,   , then the line passes through the origin O. Example 1 (a) Find a vector equation of the line 1l which is parallel to the vector 23−+i j k and passes through the point ( )5, 2, 4− . (b) Find a vector equation of the line 2l passing through points ( )5, 2, 6A and ( )3, 6, 2B − . (c) Determine if the point ( )7, 3, 7− lies on the lines 1l and/or 2l . Solution (a) 1 : , l         = +           r O A a b Here, r is the position vector of any point on the line l.

6B Vectors II 5 (b) AB OB OA=− 35 6 2 4 2 6 −               = − = =                      2 2 : 1 , 1 l  −       = +        −    r . (c) If 1 = in the equation for 1l , then 5 2 4            − + =                Therefore, the point ( )7, 3, 7− lies on 1l . Let  =  r in the equation of 2l , we have 2 1 1  −           =+           −      . As there is no solution for , the point ( )7, 3, 7− does not lie on 2l . ■ Extra Practice Annex A: Practice Question 1 1.2 Parametric and Cartesian Equations of a Line Suppose we have a line that passes through the point ( )2, 1− and is parallel to the vector 2 4   . The vector equation of this line is 22 14    =+   −    r ,   . Since r is the position vector of a point on the line, we can also write x y =  r , where ( ),xy are the coordinates of a point on the line. a b O x y r

ACJC 2025/26 H2 Mathematics (9758) 6 Since 22 14    =+   −    r , we have 22 14 x y   +   =   −+    , so 2 2 , 1 4 . x y   =+ = − + This is called the parametric equation of the line, since x and y are both given in terms of a third variable . We can make  the subject in both equations, to get 21,24 xy −+== . Therefore 21 24 xy−+ = . This is the Cartesian equation of the line. By making y the subject it can also be rearranged into the more familiar form 25yx=− . Note that we have three ways of expressing the relationship between the coordinates of the points on the line, namely the vector equation, the parametric equation, and the cartesian equation. Since the vector equation of a line is not unique, neither are the parametric equation and the cartesian equation. We can perform the same sequence of operations in three dimensions: Example Equation of line l which passes through the point ( )5, 2, 4− and parallel to 2 1 3  − can be represented in the following ways: Vector Equation 52 21 43         = − + −          r , where   . Parametric Equation Let x y z  =  r , then 52 2 43 x y z    =+ = − − =+ , where   . Cartesian Equation Make  the subject for each of the parametric equations, i.e., 5 2 4 2 1 3 x y z− + −== − .

6B Vectors II 7 In general, if we have a vector equation of a line given by =+r a b ,   , where 1 2 3 a a a  =  a and 1 2 3 b b b  =  b , to convert this to the parametric and cartesian equations, we let x y z  =  r . Then 1 1 1 1 2 2 2 2 3 3 3 3 312 1 2 3 . a b x a bx y a b y a b z a b z a b zax a y a b b b     =+         = +  = +          =+      −−− = = = Example 2 (a) Find cartesian and parametric equations of the line passing through the points ( )1, 2, 3A − and ( )4, 5, 6B − . (b) Convert the following cartesian equations of the lines to