Learning Outcomes: (a) Find the equation of a straight line. (b) Determine the angle between two straight lines. i. Vector form ii. Cartesian form The equation can be writen in (c) Find the equation of a plane. i. two planes ii. a line and a plane (d) Determine the angle between: (e) Determine the point of intersection between a line and a plane. Intersection between 2 lines Intersection between 2 planes Exclude: At the end of the lesson, students should be able to 5.4 Applications of Vectors in Geometry C05 - SM025 Page 1
Consider a line passing through point in the direction . ▪ be an arbitary point on line . be the position vector of point . [ ] ▪ be the position vector of point . [ ] ▪ Let where Note that is parallel to . Substitute Compare both sides Write in term of the rest Vector Equation Parametric Equations Cartesian Equation The Equation of a Line C05 - SM025 Page 2
Find the vector equation, the parametric equations and the Cartesian the line through in the direction of . Solution : Diagram reference : https://www.geogebra.org/m/jfrx7nbd Vector Equation Parametric Equations Cartesian Equation Example 1 C05 - SM025 Page 3
. Consider 2 lines are given by be the angle between and be the angle between Let [Note that & are corresponding angles in parallel lines] From the digram above, it is found that Based on where For & If is an obtuse angle, then it can be converted to acute angle by using . are perpendicular if [parallel in same direction] [parallel in opposite direction] are parallel if etiher one of the following is satisfied: Remarks: Angle between 2 Lines C05 - SM025 Page 4
The Cartesian equations of two straight lines and are given by Find the acute angle between the two lines. Solution : Diagram reference : https://www.geogebra.org/3d/mamh9kry be the angle between the vectors and The acute angle between the two lines is Example 2 * C05 - SM025 Page 5
Given two straight lines as follows: Show that and Solution : Since , therefore and are perpendicular. [Shown] Example 3 C05 - SM025 Page 6
Consider a plane passes through point and has a normal vector . Diagram reference: https://www.geogebra.org/3d/qtr7dd99 ▪ be an arbitary point on plane . be the position vector of point . [ ▪ ] be the position vector of point . [ ▪ ] Let Note that is perpendicular to . where is a scalar such that Substitute Vector Equation where Cartesian Equation The Equation of a Plane C05 - SM025 Page 7
Find the vector equation of a plane which contains the point and perpendicular to the line Solution : Diagram reference : https://www.geogebra.org/3d/jarzuxck Since , consider Vector Equation Example 1 C05 - SM025 Page 8
Find the vector equation and the Cartesian equation of a plane which contains the points , and Solution : Diagram reference : https://www.geogebra.org/3d/mvmyzsry Vector Equation Cartesian Equation Example 2 C05 - SM025 Page 9
. Find the vector equation of a plane which contains the lines : Solution : Diagram reference :https://www.geogebra.org/3d/x82mzeus Since both and is on the plane, either one of from and from can be chosen as in the plane equation. Let Vector Equation Example 3 C05 - SM025 Page 10
Consider 2 planes are given by be the angle between the planes and be the angle between Let Diagram reference: https://www.geogebra.org/3d/sznwrmsb From the digram above, it is found that Based on where Angle between and If is an obtuse angle, then it can be converted to acute angle by using . Remarks: Angle between 2 Planes C05 - SM025 Page 11
. Find the angle between 2 planes defined as below: Solution : Diagram reference : https://www.geogebra.org/3d/tf5wftk2 be the angle between the vectors and The acute angle between the two planes is Example 4 * C05 - SM025 Page 12
The point of intersection between a line and a plane can be obtained by: Method 1 Method 2 Vector Equation of Line Vector Equation of Plane Substitute (1) into (2) Solve & obtain Substitute into (1) Obtained a position vector & then state the point Parametric Equations of Line Cartesian Equation of Plane Substitute (1) into (2) Solve & obtain Substitute into (1) Obtained the values of & then state the point E.g. (Method 1) Vector Equation of Line ----- (1) Vector Equation of Plane ----- (2) Substitute (1) into (2) Substitute into (1) Point of intersection : E.g. (Method 2) Parametric Equations of Line , , ----- (1) Cartesian Equation of Plane ----- (2) Substitute (1) into (2) Substitute into (1) Point of intersection : Point of Intersection between a Line and a Plane. C05 - SM025 Page 13
Line : Plane : . Given If the line and the plane intersect at point , find the position vector of point . Solution : Diagram reference :https://www.geogebra.org/3d/kmtxvb7n Consider the Cartesian equation for plane & Parametric equations for line. Plane Equation Line Equation Substitute (2) into (1) From (2) Point The position vector of point is Example 5 [Example 5(a) in PowerPoint Notes Topic 5.4] C05 - SM025 Page 14
Consider a line and a plane are given by be the angle between and be the angle between Let Diagram reference: https://www.geogebra.org/3d/xw4wt2mm From the digram above, note that is an acute angle & Remarks: If is an obtuse angle, then Based on where For and Remarks: If is an obtuse angle, then the angle between is Angle between a Line and a Plane C05 - SM025 Page 15
Line : Plane : . Given Find the angle between the line and the plane . Solution : Diagram reference : https://www.geogebra.org/3d/epfqy5wx be the angle between the vectors and The acute angle between the line and the plane is Example 6 [Example 5(b) in PowerPoint Notes Topic 5.4] C05 - SM025 Page 16
Find the parametric equations for the line passing through the point and . (a) (b) Where does the line in part (a) intersects the plane ? Solution (a) : Diagram reference : https://www.geogebra.org/3d/vgjyvf23 Since both and is on the line , either one of and can be chosen as in line equation. Let Parametric Equations Solution (b) : plane When The line intersects the plane at the point Example 7 C05 - SM025 Page 17