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**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

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**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

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**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

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**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

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**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."

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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

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**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) , ) p=(-2,5,-1) ,

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**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

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**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

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**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)

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**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

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**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

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**5. If a line L has parametric equations**

find the plane contain L and point P=(5,0,2)

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**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

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**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

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**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

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**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,

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**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

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**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.

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**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

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**Finding a common perpendicular**

Read section 13.4 p 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:

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