Planes in three dimensions Normal equation Cartesian equation of plane Angle between a line and a plane or between two planes Determine whether a line intersects or lies in or parallel to a plane Distance from a point to a plane

Normal equation The normal is perpendicular to any line in the plane n (p,q,r) n (x,y,z) R A (a,b,c) Cartesian equation of a plane o

theorems The graph of every linear equation is a plane with normal vector (a,b,c) 2. Two planes with normal vectors a & b are (i) parallel if a and b are parallel (ii) orthogonal if a and b are orthogonal

Extra :Cross product Definition: is right-handed where determinant where x,y,and z are unit vectors, here u x v is always perpendicular to u and v Therefore we can find the normal of the plane from two vectors on the plane

Cross product 1. The answer is "Elephant grape sine-theta." The magnitude of the cross product is given by What’s the dot product??? Joke presented on the television sitcom Head of the Class . The teacher asks: "What do you get when you cross an elephant and a grape?" The answer is "Elephant grape sine-theta."

Examples Find an equation of the plane through the point (5,-2,4) with normal vector a=(1,2,3) Prove that the planes and are parallel

1) through P (-2,5,-8) with normal vector a=(-1,-4,1) 3. Find an equation of the plane which satisfies the stated conditions: 1) through P (-2,5,-8) with normal vector a=(-1,-4,1) 2) through P (2,5,-6) and parallel to the plane 3) through the points P(3,2,1) ,Q(-1,1,2) ,R(3,-4,1) 4. find the distance from the point P to the given plane 1) p=(2,1,-1) , 2) p=(-2,5,-1) ,

Angle ; projection of a line on a plane 1. Projection : L Normal n AB is the projection of line L on the plane B A projection 2. Angle between a line and the plane The angle between line L and its projection on the plane

Angle between two planes 1. First find the angle between two normal to the planes n2 n1 From dot product to find the angle theta 2. The angle between two planes is

Find the common perpendicular 1*.cross product uxv is perpendicular to both u and v. For example: find the common perpendicular to u=(1,2,3) and v=(7,8,9)

Find the common perpendicular vector Technique introduced in the textbook. the common perpendicular vector of Beware of the second term: the order of subtrahend is different from the other and is

Find the normal to the plane Procedure: 1.Find the two vectors lie on the plane 2. find cross product of the two vectors Example: find the Cartesian equation of the plane through A(1,2,1), B(2,-1,-4) and C(1,0,-1) Two vectors on the plane: AB=(-1,3,5) and AC=(0,2,2) The vector is perpendicular to both AB and AC Therefore -4i+2j-2k is one of the normal to the plane ,equation is -4x+2y-2z=-2

5. If a line L has parametric equations find the plane contain L and point P=(5,0,2)

1 2 4 3 Line A Line A Line B Point B Angle between lines Skew Find the perp. distance of the point from the line Intersect B is on the line A Parallel (- identical?) Point A Plane B 4 3 Line A Plane B A is in plane Line intersects the plane at one point Line is in the plane B Find the perp. distance of the point from the plane A is parallel to B

5 1 Plane A Plane B Angle between planes There is a line of Intersection m Parallel (- identical?) 1 Given the two vector equations of the lines, you can examine the three simultaneous equations: If two equations yield a specific pair of values (t,s) and the third equation is consistent, then there is an intersection. If there are no solutions, the lines are skew or parallel. If q is a multiple of p, the lines are parallel, otherwise they are skew. If there are an infinite number of solutions, the lines are identical. p θ The angle between two lines is easily found using the dot product. q

2 To examine whether a point is on a line, simply find a value of the Parameter t which satisfies the equation for each coordinate: If there is no value of t which makes these equations all true, then the point m is not on the line. To find the distance from the point m to the line: calculate the unit normal vector Find the displacement vector from A to m Take the dot product of these two vectors Use Pythagoras to find the third side of the triangle X d A M

3 Line A Plane B Given a plane: and a line: … the line is parallel to the plane if: Moreover it is in the plane, also, if: This must be true for all t, so t must cancel out and the LHS=RHS= a constant. If there are no solutions for t: the line is parallel to the plane, but not in it. If there is one solution for t: the line intersects the plane. In the last case,

4 n Point A Given a plane: Plane B and a point: There are only two possibilities: the point is in the plane, or it isn’t. To check if it is in the plane: simply substitute the coordinates of m into the equation of the plane. n To find the distance of the point from the plane: A (a,b,c) The simplest way is to find a unit normal vector and then calculate: d m

5 Plane A Given a plane: Plane B and a plane: The planes may be parallel or identical or …? Intersect in a line n2 n1 This diagram shows how to find the angle between the planes.

5 Plane A Given a plane: Plane B and a plane: To find the equation of the line: Choose any two values of x. Use the two plane equations to find the corresponding y and z which solves both plane equations Now you have two points on the line, so you can find its equation

Finding a common perpendicular Read section 13.4 p. 182-3 carefully. The method described is actually the method to find the cross product. Find a vector perpendicular to each pair of vectors: Note that this method is a good way to find the equation of a plane through three points A, B, C: