2026 Chp 1C (Student) - JPJC

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2026 Chapter 1(c) Vectors − Planes (Students’ version) / Pg 1 Chapter 1(c) : Planes 1. Equation of a Plane A plane is commonly defined using (i) two vectors parallel to the plane and a point on the plane or (ii) three non-collinear points or (iii) the direction of the normal to the plane and one point on the plane. In this section, we will learn the three different forms of equation of a plane, namely 1.1 Vector equation (in parametric form): = + +r a b c , where ,   ; 1.2 Vector equation (in scalar product form): d•=rn , where d   ; and 1.3 Cartesian equation: ax by cz d+ + = , where , , , a b c d   . Recall: An equation of a line is of the form r = a + b ,    where r represents position vector of a point on the line a represents position vector of a given (known) point on the line b represents a vector parallel to the line 1.1 Vector Equation of a Plane in Parametric Form Refer to the diagrams below. A is a fixed point on a plane and R1, R2, R3, ……etc are different points on the same plane. b and c are 2 vectors parallel to the plane. Then, by vector addition, 1 1 1AR =+ bc  2 2 2AR =+ bc  3 3 3AR =+ bc  Generalising, for any point R lying on the plane, AR =+ bc  , where ,   . In fact, any vectors on the plane or parallel to the plane can also be expressed as  +bc . c b A R2 R3 c b A c b A R1

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2026 Chapter 1(c) Vectors − Planes (Students’ version) / Pg 2 We can obtain the vector equation of a plane in parametric form for the following cases: ➢ Given a point on the plane and two vectors parallel to the plane The equation of the plane through any given point A, with position vector a, and parallel to vectors b and c can be found as follows. OR OA AR=+  where R is any variable point on the plane.  = + +r a b c , ,   Vector equation of the plane in parametric form: = + +r a b c , ,   Example 1 Write down the vector equation of the plane in parametric form (a) through point C(4, –1, 2) and parallel to the vectors i + 3j + k and –i + 4j – k, (b) through the points A(3, –2, 0), B(2, 0, 3) and C(1, −1, 1) (c) that contains the lines ( ) ( ) and 2ts= − + + = − + − −r j k i k r j k i j k Solution (a) 4 1 1 1 3 4 2 1 1  −           = − + +           −      r , ,   (b) OA AC CB= + +r  3 2 1 2 1 1 0 1 2  −           = − + +                r , ,   (c) 0 1 2 1 0 1 1 1 1             = + + −          −−      r , ,   O c b A R O R C O R O R

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