Date Lesson TOPIC Homework. The Intersection of a Line with a Plane and the Intersection of Two Lines

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1 UNIT 4 - RELATIONSHIPS BETWEEN LINES AND PLANES Date Lesson TOPIC Homework Oct The Intersection of a Line with a Plane and the Intersection of Two Lines Pg. 496 # (4, 5)b, 7, 8b, 9bd, Oct Systems of Equations Pg. 507 # (3, 5, 6)a, 9, Oct Intersection of Two Planes Pg. 56 # a, a, 5, 6, 0 Oct Intersection of Three Planes Pg. 530 #,, 4, 8ad, 9bc, 3af I.S Distance from Point to Line & Point to Plane Pg. 540 # (,, 3, 5)a, 6b, 9 Pg. 550 #, ace, 3, 5 Oct Review for Unit 4 Test Pg. 55 # -3, 6, 8, 0,, 4, 9, Oct TEST - UNIT 4 Oct. 3-6 Review for Vectors Summative Pg 557 # -, 6a, 7-9a, 0,, 3-9, 3-34 Oct. 7 Vectors Summative

2 MCV 4U Lesson 4. The Intersection of a Line and a Plane and The Intersection of Two Lines INTERSECTION of a LINE and a PLANE Zero solutions One solution Infinite # of solutions Note: If a line is parallel to the plane, then the dot product of a normal to the plane and the direction vector of the line will be zero. Methods to solve a system: Write each equation in parametric form and solve for each variable. OR If a scalar equation is given, directly substitute each component into the equation. Ex. Determine the intersection of: 4x 5y 4z x 5 y 4 z t 3t t ----

3 Ex. Determine the intersection of: 3x 9y 7z x y z Ex. 3 Determine, without solving, if each line intersects the plane. a) l : r (, 5,3) s(3,,) b) l : r (,0,) t(,,4) : 3x y z 6 : 4x z

4 INTERSECTION of TWO LINES Recall: A linear equation can be written in the form a x a x a 3 x 3... a n x n k, where a,... and k are constants and x,... are variables. A system of linear equations may have: a) Exactly one solution b) No Solutions c) An infinite number of solutions In 3 space, lines that that are neither intersecting nor parallel are said to be "skew". If a system has at least one solution, the system is said to be "consistent". A system is "inconsistent" if it has no solutions. Methods to solve a system of equations include substitution and elimination.

5 Ex. 4 Find the intersection of the lines. a) x 3t x 7s r r b) y 4t y 5s r r (7,, 6) s(,, 3) (3,9,3) t(,5,5) Pg. 496 # (4, 5)b, 7, 8b, 9bd,

6 MCV 4U Lesson 4. Systems of Equations Number of Solutions to a Linear System of Equations A linear system of equations can have zero, one or an infinite number of solutions. Definition of Equivalent Systems Two systems of equations are defined as equivalent if every solution to one system is also a solution to the second system of equations Elementary Operations Used to Create Equivalent Systems of Equations. Multiply an equation by a nonzero constant.. Interchange any pair of equations. 3. Add a nonzero multiple of one of one equation to a second equation to replace the second equation Consistent and Inconsistent Systems of Equations A system of equations is consistent if it has either one solution or an infinite number of solutions. A system of equations is inconsistent if it has no solution. Ex. Solve. 5x y 7z x y z x 4y z ---

7 Ex. 3 Determine the value of k for which the system below has : x y --- x k y k --- a) no solutions b) solution c) infinite solutions Pg. 507 # (3, 5, 6)a, 9,

MCV 4U Lesson 4.3 Intersection of Two Planes Given two planes in three space, there are three possible geometric models for the intersection of the planes.

8 MCV 4U Lesson 4.3 Intersection of Two Planes Given two planes in three space, there are three possible geometric models for the intersection of the planes. If the planes are parallel and distinct, they do not intersect and there is no solution. If the planes are coincident, every point on the plane is a solution. If two distinct planes intersect, the solution is the set of points that lie on the line of intersection. Ex. Describe how the planes in each pair intersect and if they intersect, find the solution. a) : x y z 0 : x y z 6 0 When determining if planes are parallel for the purpose of determining the intersection of planes it is useful to include the constant to determine if parallel planes are distinct or coincident. Write normals in simplest form. Write as (A, B, C); D

9 b) 3 : x 6 y 4z : 3x 9 y 6z 0 c) 5 : x y z 0 6 : x y 4z 4 0 Ex. Describe how the planes intersect. r (,4,9) s(3,,4) t(,, ) r (4,8,) m(4,,) n(3,,) Pg. 56 # a, a, 5, 6, 0

MCV 4U Lesson 4.4 Intersection of Three Planes A system of three planes is consistent if it has one or more solutions. The planes intersect at a point. The planes intersect in a line.

The normals are coplanar, of solutions. The normals are coplanar. but not parallel. parallel and coplanar. A system of three planes is inconsistent if it has no solution.

The third plane is not Pairs of planes intersect in normals are parallel. parallel. Two of the normals are lines that are parallel and parallel. distinct. The normals are coplanar but not parallel.

10 MCV 4U Lesson 4.4 Intersection of Three Planes A system of three planes is consistent if it has one or more solutions. The planes intersect at a point. The planes intersect in a line. The planes are coincident. There is exactly one solution. There are an infinite number of There are an infinite number The normals are NOT parallel and not solutions. The normals are coplanar, of solutions. The normals are coplanar. but not parallel. parallel and coplanar. A system of three planes is inconsistent if it has no solution. The three planes are parallel and Two planes are parallel and distinct The planes intersect in pairs. at least two are distinct. The distinct. The third plane is not Pairs of planes intersect in normals are parallel. parallel. Two of the normals are lines that are parallel and parallel. distinct. The normals are coplanar but not parallel. It is easy to check if normals are parallel; each one is a scalar multiple of the others. To check if normals are coplanar, use the triple scalar product n (n n 3 ). Remember that this product gives us the volume of a parallelepiped defined by the three vectors. If the product is zero, the volume is zero and the vectors must be coplanar. If the product is not zero the vectors are not coplanar.

11 Ex. Determine the intersection for each set of planes. a) b) : x y 6z : x 5y z 0 0 : 3x 4y 3z : x 7 y z : x y 4z : 8x 5y z 0 0

12 Ex. Determine if each system can be solved; then solve the system or describe it. a) 3x + y z = b) x + 3y z = 0 c) 4x y + 6z = 35 x 5y + z = 8 x + y + z = 8 0x + 5y 5z = 0 x + 4y 8z = 4 x y + z = 4 6x 3y + 9z = 50

13 One Point of Intersection One Line of Intersection Source: Triangular Prism No Intersection Two Parallel Planes No Intersection Three Parallel Planes No Intersection Pg. 530 #,, 4, 8ad, 9bc, 3af

14 MCV 4U Independent Study 4.5 Distance from a Point to a Line in R and R 3 DISTANCE from a POINT to a LINE in R To find the distance from a point Q(x, y ) to a line with scalar equation Ax By C 0, we can let a point on the line be P( x o, y o ) and the distance be d. d = PQ = proj( PQ) on to n Q n PQ n n R = ( x x o, y y o ) ( A,B) A B P = Ax Ax o By By o A B, (x o, y o ) is a point on the line Ax o and By o are constants. Ax o By o C So, d = Ax By C A B The distance from a point (x, y ) to the line Ax By C 0 is given by the formula d = Ax By C A B Ex. Find the distance between the lines x 3y + = 0 and x + 3y 5 = 0. n (, 3) n (, 3) parallel - need only find a point on one line and find distance from that point to the other line. For x 3y 0, point on line (0, 4) find distance from (0, 4) to x + 3y 5 = 0 d Ax By A B C (0) 3(4) 5 () (3)

15 DISTANCE from a POINT to a LINE in R 3 P r ro sm, sr d R T d = distance b/w P and the line P = a point that is known Q = any point on the line whose coords are known T = point on line such that QT is a vector representing the direction m, which is known m Q In PQR, sin = d = d QP QP sin from cross product, we know m QP m QP sin If we substitute d = QP sin into this formula, we find that m QP m QP m (d) Solving for d gives d = m The distance from a point P ( x, y, z) to the line r ro sm, sr, in R 3, is given by the formula m QP d =, where Q is a point on the line and P is any other point, both of whose coordinates m are known, and m is the direction vector of the line. Ex. Find the distance between the point A(, -3, 5) and the line r (0,5,) s(4,,3 ), sr Q ( 0, 5, ) P (, 3, 5) QP (, 8, 3) From the equation, we know m = (4, -, 3) d m QP m (4,, 3) (, 3, 5) (4,, 3) (4, 4, 0) (4) ( ) (3) (4) ( 4) 6 ( 0) Pg. 540 # (,, 3, 5)a, 6b, 9

16 MCV 4U Independent Study 4.6 Distance from a Point to a Plane Distance from a Point to a Plane If there is a point Q x, y, ) off the plane and a point P x, y, z ) on the plane Ax By Cz D 0, ( z o ( o o o then the distance d from Q to the plane is the projection of P o Q onto the normal ( A, B, C). d proj( P Q) onto n o P o Q n n ( x xo, y yo, z zo ) ( A, B, C) A B C A( x xo ) B( y yo ) C( z zo ) A B C Ax By Cz ( Axo By o Czo ) A B C Since P is a point on the plane, it satisfies the plane, so Ax By Cz D 0 o Substituting this into the above equation gives d Ax By A D Ax By Cz. o o o or o o o B Cz D C The distance from a point (x, y, z ) to the plane Ax By Cz D 0 is given by the formula Ax By Cz D Po Q n d OR d, where n is the normal vector of the plane A B C n and P o is a point on the plane and Q is the point whose coordinates are known. Ex. Find the distance from the point Q (0, 3, -8) to the plane 4x y z 6 0. Point on the plane, P o = (4, 0, 0) P Q ( 6, 3, 8) o P Q n ( 6, 3, 8) (4,) o d P Q n o n (4) () () 4.80 OR d Ax By A 4(0) (3) ( 8) B Cz C D

17 Distance between Skew Lines Skew lines are lines in 3 space that are not parallel and do not intersect. Even though they are not parallel, they do not intersect because they lie in different planes. They pass each other just like vapour trails left by two aircraft flying at different altitudes. Ex. Determine the distance between L : r (,4, 5) s(,0,3 ) and L : r (,, ) t(,,0 ). To find the distance between skew lines ( lines which do not intersect and are not parallel), we need a point on both lines ( P and P ), the vector P P, and the normal, n, to both lines. The distance is equal to the scalar projection of P P onto n. Recall: Projection of P P onto n P P n n P (, 4, 5) n d d P (,, ) P P ( 4,, 4) Proj n P P ( 4,, 4) ( 3, 6,) ( 3) OR n ( 3, 6,) 3x 6y z D 0 point on L (, 4, 5) 3() 6(4) ( 5) D 0 D 3 L is 3x 6y z 3 0 point on L (,, ) Ax d By A B Cz 3( ) 6() ( ) 3 ( 3) (6) () C D Pg. 550 #, ace, 3, 5

The Three Dimensional Coordinate System

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