Scalar Equation of a Plane MCV4U: Calculus & Vectors Imagine a - PDF document

p l a n e s p l a n e s Scalar Equation of a Plane MCV4U: Calculus & Vectors Imagine a plane containing point P ( x p , y p , z p ), which is known, and a general point Q ( x q , y q , z q ). Scalar Equation of a Plane J. Garvin The vector � PQ = ( x q − x p , y q − y p , z q − z p ) represents a vector in the plane. Let � n = ( A , B , C ) be a known normal to the plane. J. Garvin — Scalar Equation of a Plane Slide 1/16 Slide 2/16 p l a n e s p l a n e s Scalar Equation of a Plane Scalar Equation of a Plane n · � According to the dot product, � PQ = 0. Example Determine the scalar equation of the plane with normal ( A , B , C ) · ( x q − x p , y q − y p , z q − z p ) = 0 � n = (3 , − 2 , 5) that contains the point P (1 , 0 , − 1). Ax q − Ax p + By q − By p + Cz q − Cz p = 0 Ax q + By q + Cz q − ( Ax p + By p + Cz p ) = 0 3(1) − 2(0) + 5( − 1) + D = 0 Since all values in − ( Ax p + By p + Cz p ) are known, replace it with a constant D . − 2 + D = 0 Scalar Equation of a Plane D = 2 The scalar equation of a plane, with normal vector The scalar equation is 3 x − 2 y + 5 z + 2 = 0. n = ( A , B , C ), is Ax + By + Cz + D = 0. � J. Garvin — Scalar Equation of a Plane J. Garvin — Scalar Equation of a Plane Slide 3/16 Slide 4/16 p l a n e s p l a n e s Scalar Equation of a Plane Scalar Equation of a Plane Example Use � n and a point in the plane to find the scalar equation. Determine the scalar equation of the plane containing the 10(1) − 2(0) + 7(3) + D = 0 points P (1 , 0 , 3), Q (2 , − 2 , 1) and R (4 , 1 , − 1). 31 + D = 0 To find the scalar equation, we need to calculate a normal to D = − 31 the plane. The scalar equation is 10 x − 2 y + 7 z − 31 = 0. Two vectors in the plane are � PQ = (1 , − 2 , − 2) and � QR = (2 , 3 , − 2). The cross product can be used to find a vector that is perpendicular to any two vectors contained in the plane. n = (1 , − 2 , − 2) × (2 , 3 , − 2) = (10 , − 2 , 7). � J. Garvin — Scalar Equation of a Plane J. Garvin — Scalar Equation of a Plane Slide 5/16 Slide 6/16

p l a n e s p l a n e s Scalar Equation of a Plane Scalar Equation of a Plane Example Example Determine the scalar equation of the plane with vector Determine the vector and parametric equations of the plane equation � r = (3 , 1 , − 2) + s (1 , 0 , 2) + t ( − 2 , 1 , 0). with scalar equation 3 x + 2 y − z − 5 = 0. Two direction vectors are given in the vector equation, so A simple method is to find three points on the plane, and use find the cross product of them to determine the normal to these points to create the two required direction vectors. the plane. Begin by rewriting the equation of the plane as n = (1 , 0 , 2) × ( − 2 , 1 , 0) = ( − 2 , − 4 , 1). z = 3 x + 2 y − 5. � Use the normal and a point on the line to determine the x y z = 3 x + 2 y − 5 Point scalar equation. 0 0 z = 3(0) + 2(0) − 5 = − 5 P (0 , 0 , − 5) − 2(3) − 4(1) + 1( − 2) + D = 0 0 1 z = 3(0) + 2(1) − 5 = − 3 Q (0 , 1 , − 3) D = 12 1 0 z = 3(1) + 2(0) − 5 = − 2 R (1 , 0 , − 2) The scalar equation is 2 x + 4 y − 1 z − 12 = 0. J. Garvin — Scalar Equation of a Plane J. Garvin — Scalar Equation of a Plane Slide 7/16 Slide 8/16 p l a n e s p l a n e s Scalar Equation of a Plane Scalar Equation of a Plane Two direction vectors for the line are � PQ = (0 , 1 , 2) and If two planes are parallel (or coincident), their normals will � QR = (1 , − 1 , 1). also be parallel. A possible vector equation for the line is This implies that � n 1 = k � n 2 for some real value k . � r = (1 , 0 , − 2) + s (0 , 1 , 2) + t (1 , − 1 , 1). If two planes are perpendicular, their normals will also be The corresponding parametric equations are x = 1 + t , perpendicular. y = s − t and z = − 2 + 2 s + t . This implies that � n 1 · � n 2 = 0. J. Garvin — Scalar Equation of a Plane J. Garvin — Scalar Equation of a Plane Slide 9/16 Slide 10/16 p l a n e s p l a n e s Scalar Equation of a Plane Scalar Equation of a Plane Example Example Determine if the planes π 1 : x − 3 y + 5 z + 4 = 0 and Determine if the planes π 1 : x − 3 y + 5 z + 4 = 0 and π 2 : 4 x − 12 y + 20 z + 18 = 0 are coincident, parallel, π 2 : x + 2 y + z + 3 = 0 are coincident, parallel, perpendicular perpendicular or neither. or neither. From the given equations, � n 1 = (1 , − 3 , 5) and � n 2 = (1 , 2 , 1). From the given equations, � n 1 = (1 , − 3 , 5) and n 2 = (4 , − 12 , 20). Since � � n 2 = 4 � n 1 , the planes are either Using the dot product, (1 , − 3 , 5) · (1 , 2 , 1) = 0. coincident or parallel. Therefore, planes π 1 and π 2 are perpendicular. Since the equation for π 2 is not a scalar multiple of the equation for π 1 (the constant is off), the planes are not coincident. Therefore, planes π 1 and π 2 are parallel. J. Garvin — Scalar Equation of a Plane J. Garvin — Scalar Equation of a Plane Slide 11/16 Slide 12/16

p l a n e s p l a n e s Scalar Equation of a Plane Scalar Equation of a Plane Example Two non-parallel planes must intersect (more on this later). Determine the angle formed between the intersecting planes The angle formed between the two planes will be the same as π 1 : x − y − 2 z + 3 = 0 and π 2 : 2 x + y − z + 2 = 0. the angle formed between their normals. We can use the dot product to calculate the measure of this From the given equations, � n 1 = (1 , − 1 , − 2) and angle. n 2 = (2 , 1 , − 1). � Angle Between Two Intersecting Planes Calculate the magnitudes of � n 1 and � n 2 . The angle, θ , between two intersecting planes π 1 and π 2 is � 1 2 + ( − 1) 2 + ( − 2) 2 | � n 1 | = given by cos θ = � n 1 · � n 2 n 2 | . √ | � n 1 || � = 6 � 2 2 + 1 2 + ( − 1) 2 | � n 2 | = √ = 6 J. Garvin — Scalar Equation of a Plane J. Garvin — Scalar Equation of a Plane Slide 13/16 Slide 14/16 p l a n e s p l a n e s Scalar Equation of a Plane Questions? Use the dot product to calculate the angle between the normals and, thus, between the planes. cos θ = (1 , − 1 , − 2) · (2 , 1 , − 1) √ √ 6 6 = 2 − 1 + 2 6 = 1 2 Therefore, the angle between the planes is 60 ◦ (or 120 ◦ ). J. Garvin — Scalar Equation of a Plane J. Garvin — Scalar Equation of a Plane Slide 15/16 Slide 16/16