6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line

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1 CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle between Two Planes 6..3 Angle between a Line and a Plane 6..4 Shortest Distance from a Point to a Plane Reiew: Basic Concepts Vectors in Space The Dot Products The Cross Products

2 Basic Concepts What is scalar? a quantity that has only magnitude What is ector? a quantity that has magnitude and direction A ector can be represented by a directed line segment where length of the line segment - the magnitude of the ector direction of the line segment - the direction of the ector

3 Q x, y Terminal Point PQ P x y, Initial Point A ector can be written as PQ, or a. The order of the letters is important. PQ means the ector is from P to Q or the position ector Q relatie to P, QP means ector is from Q to P or the position ector P relatie to Q. If P x is the initial point and Q, y x, y is the terminal point of a directed line segment, PQ then component form of ector that represents PQ is 3

4 x, x x, y y x y x y and the magnitude or the length of is x x y y c c P x y, Q x, y OP x 0, y 0 x, y OQ x 0, y 0 x, y -Note- Any ector that has magnitude of unit = unit ector. 4

5 Example: Find the component form and length of the ector that has initial point 3, 7 and terminal point Solution:,5. 3,5 7 5, Example: Gien, 5 and w 3, 4 the following ectors:, find each of a) Answer: b) w c) w a),5 b) 5, c) 4,3 5

6 Theorem: If a is a non-null ector and if â is a unit ector haing the same direction as a, then a a= ˆ a -Note- To erify that magnitude is, â = Example: Find a unit ector in the direction of,5 and erify that it has length. Solution:,5,

7 Standard Unit Vectors Three standard unit ectors are: i, j dan k z ˆk î ĵ y x Vectors i, j and k can be written in components form: i = <, 0, 0 >, j = < 0,, 0 > and k = < 0, 0, > and can interpreted as a = <x, y, z> = xi + yj + zk 7

8 The ector PQ with initial point Px, y z and terminal point Q x, y z, has the standard representation PQ Example:, ( x x z ) i ( y y ) j ( z ) k or PQ x x, y y, z z Let u be the ector with initial point (,-5) and terminal point (-,3), and let i j. Write each of the following ectors as a linear combination of i and j. a) u b) w u 3 8

9 y y 0 x x -Note- If is the angle between and the positie x axis then we can write x cos and y sin ; x y. Example: The ector has a length of 3 and makes an 30 angle of 6 with the positie x-axis. Write as a linear combination of the unit ectors i and j. 9

10 Vectors in Space Properties of Vectors in Space Let,, 3 and w, w, w3 w be ectors in 3 dimensional space and k is a constant.. w if and only if w w, 3 w3,.. The magnitude of is 3 3. The unit ector in the direction of is,, 3 4. w w, w, 3 w3 5. k k, k, k3 6. Zero ector is denoted as 0 0, 0, w w. 0

11 8. Let u u, u, u3, 9. u+ 0 then + + u 0. u+ -u = 0. c +d u = cu+du. + c u + cu c 3. cdu cd u + w u w. u 4. u = u and 0u = 0 5. c u =c u Example: Express the ector PQ if it starts at point P ( 6, 5, 8) and stops at point ( 7, 3, 9) components form. Q in

12 Solution: PQ x x, y y, z z PQ 7 6,3 5,9 8 PQ,, Example: Gien that 3,, a, b, 6, 4. Find (a) a 3b (b) b (c) a unit ector which is in the direction of b. (d) find the unit ector which has the same direction as Answer: a 3b. (a) 0,9,0 (b) 53 (c) 4 3,, 6 3 (d) (e),6,4 53 (f) 46 0,9,0

13 Parallel Vector hae same slopes ; constants So, there are multiples of each other. Example: Vector w has initial point (,-,3) and terminal point (-4,7,5). Which of the following ectors is parallel to w? Solution: w 4,7,5 3 w 6,8, One example: 6,8,,6,4 Another is 6,8, 3,4,. 3

14 Example: (Collinear Points) Determine whether the point P,,3, Q,,0 and R 4,7, 6 lie on the same line. Solution: PQ,,0 3,3, 3 PQ QR 4,7, 6 0,6, 6 QR PR 4,7, 6 3 3,9, 9 PR Thus, PR PQ QR. Since one ector is a multiple of the other, the two ectors are 4

15 parallel and since they share a common point Q, they must be the same line. The Dot Product -Theorem- If,, 3 and w w, w, w3, then the scalar product -Note- w is w, 3, w, w, w3 w w 3w3 The dot product is also called - the scalar product - the inner product The dot product of two ectors is a scalar. 5

16 The Angle between Vectors Refer to the figure below, let,, 3 u u OP u u u,,, 3 OQ be two ectors and let be the angle between them, with 0. z Q,, 3 c P u, u, u 3 u y x Compute the distance, c between points P and Q in two ways. 6

17 ) Using the Distance formula c u u u3 3 u u u 3 3 u u u 3 3 u u u u33 ---() ) Using the Law of Cosines c u u cos ---() Equating equation () and (), we get cos u u u33 u u u Example: If = i-j+k, w = i+j+k and the angle between and w is 60, find w. 7

18 Solution: w cos cos Example: Gien that,, 3 u, 5,8, and w 4,3,, find (a) (b) u u w (c) u w (d) the angle between u and (e) the angle between and w. Answer: (a) -3 (b), 9,6 (c) -3 8

19 (d) 94 4 (e) Example: Let A=(4,,), B=(3,4,5) and C=(5,3,) are the ertices of a triangle. Find the angle at ertex A. Answer: 79 -Theorem- The nature of an angle, between two ectors u and. is an acute angle if and only if u 0 is an obtuse angle if and only if u 0 = 90 if and only if u 0. The Vectors u and are orthogonal / perpendicular. 9

20 Example: Show that the gien ectors are perpendicular to each other. (a) i and j (b) 3i-7j+k and 0i+4j-k -Theorem- (Properties of Dot Product) If u, and w are nonzero ectors and k is a scalar,. u u. u w u u w 3. ku u k u 0 0u 0 0

21 The Cross Products - The cross product (ector product) u is a ector perpendicular to u and. (illustrated in figure below) - The direction is determined by the right hand rule. u u If the first two fingers of the right hand point in the directions of and respectiely, then the thumb points in the direction of u Ex: i j k. u

22 - The length is determined by the lengths of u and and the angle between them. - If we change the order informing the cross product, then we change the direction. Ex: -Theorem- If then, u u3 u u u i u j k and i j k, 3 u i j k u u u 3 3 u u u u uu i j k u u u u u u i j k

23 Properties of Cross Product (a) u u 0 (b) u u (c) u w u u w (d) k u uk ku (e) u // if and only if u 0 (f) u 0 0u 0 Example: ) Gien that u 3,0, 4 and,5,, find (a) u (b) u ) Find two unit ectors that are perpendicular to the ectors and = i+3j+k. u i+j-3k 3

24 Answer: ) (a) -0i+0j+5k (b) 0i-0j-5k, 5, 4 ) 6 (The unit ector in the opposite direction is also a unit ector perpendicular to both u and ) Further geometry interpretation of the cross product comes from computing its magnitude. u u u u u u u u u u u u u u 3 3 u u u u u u cos sin cos 4

25 with is the angle between u and. Therefore, u u sin. B C u u sin A D From the figure aboe, we can see that the magnitude of the cross product is the area of the parallelogram of which arrows representing the two ectors are adjacent sides. Area of a parallelogram u sin u Area of triangle = u 5

26 Example: (a) Find an area of a parallelogram that is formed from ectors u = i + j - 3k and = -6j + 5k. (b) Find an area of a triangle that is formed from ectors u = i + j - 3k and = -6j + 5k. Answer: (a) 30 (b) 30 Scalar Triple Product -Theorem- a x, y, z, x, y, z If c x3, y3, z3, b and then 6

27 a b c x x x 3 y y y 3 z z z 3 Properties of The Scalar Triple Product ) a b c a bc ) abc acb bca 3) acb abc 4) aab 0 5) a dbc abc dbc Example: If a = 3i + 4j - k, b = -6j+5k and c = i+j-k, ealuate (a) a b c (b) a bc (c) c ab (d) b a c 7

28 6. Lines in Space In this section we use ectors to study lines in three-dimensional space. HOW LINES CAN BE DEFINED USING VECTORS? The most conenient way to describe a line in space is to gie a point on it and a nonzero ector parallel to it. Suppose L is a straight line that passes through P x, y, ) and is parallel to the ( 0 0 z0 ector ai bj ck. 8

29 Thus, a point Q ( x, y, z) also lies on the line if Let, PQ t. r0 OP and r OQ, Then PQ r r 0. r r 0 t r r 0 t x, y, z x0, y0, z0 t a, b, c -Theorem- (Parametric Equations for a Line) The line through the point P x, y, ) and ( 0 0 z0 parallel to the nonzero ector A a, b, c has the parametric equations, 9

30 x x 0 at, y y 0 bt, z z 0 ct. If we let R 0 x0, y0, z0 denote the position ector of P x, y, ) and R x, y, z the ( 0 0 z0 position ector of the arbitrary point Q(x,y,z) on the line, then we write equation () in the ector form, Example: R R 0 ta. Gie the parametric equations for the line through the point (6,4,3) and parallel to the ector,0, 7. -Theorem- (Symmetric Equations for a line) 30

31 The line through the point P x, y, ) and parallel to the nonzero ector ( 0 0 z0 A a, b, c has the symmetrical equations, x x0 y y0 z z0 a b c. Example: Gien that the symmetrical equations of a line in space is Find, x 3 y z (a) a point on the line. (b) a ector that is parallel to the line.. 3

32 6.. Angle between Two Lines Consider two straight lines x a x y l : b and y z x x y y z z : d e f l c z The line l parallel to the ector u ai bj ck and the line l parallel to the ector di ej fk. Since the lines l and l are parallel to the ectors u and respectiely, then the angle, between the two lines is gien by cos u u 3

33 Example: Find an acute angle between line and line l = i + j + t (i j + k) l = i j + k + s (3i 6j + k). 33

34 6.. Intersection of Two lines In three dimensional coordinates (space), two line can be in one of the three cases as shown below, 34

35 35 Let l and l are gien by: () : () and : f z z e y y d x x l c z z b y y a x x l From (), we hae c b a,, From (), we hae f e d,, Two lines are parallel if we can write The parametric equations of l and l are: ct z z bt y y at x x l : (3) : fs z z es y y ds x x l

36 Two lines are intersect if there exist unique alues of t and s such that: x y z at bt ct x y z ds Substitute the alue of t and s in (3) to get x, y es and z. The point of intersection = (x, y, z) Two lines are skewed if they are neither parallel nor intersect. Example: Determine whether l and l are parallel, intersect or skewed. a) l : x 3 3t, y 4t, z 4 7t l : x 3s, y 5 4s, z 3 7s fs 36

37 x y b) l : z 4 l : x 4 Solutions: a) for l : y 3 z 3 point on the line, P = (3,, - 4) ector that parallel to line, for l : point on the line, Q = (, 5, 3) ector that parallel to line, where Therefore, lines l and l are parallel.? 3, 4, 7 3, 4, 7 b) Symmetrical equations of l and l can be rewrite as: 37

38 l l : x y z 0 4 x 4 y 3 z ( ) : 3 Therefore: for l : P = (,, 0),, 4, for l : Q = (4, 3, -),,, 3? not parallel. In parametric eq s: l : x t, y 4 t, z t l : x 4 s, y 3 s, z 3s t 4 s () 4t 3 s () t 3 s (3) 38

39 Sole the simultaneous equations (), (), and (3) to get t and s. 5 7 s and t 4 4 The alue of t and s must satisfy (), () and (3). Clearly they are not satisfying () i.e ? Therefore, lines l and l are not intersect. This implies the lines are skewed! Example: Let L and L be the lines L : x 4t, y 5 4t, z 5t L : x 8t, y 4 3t, z 5 t (a) Are the lines parallel? (b) Do the lines intersect? 39

40 6..3 Shortest Distance from a Point to a Line Q PQ sin P V Distance from a point Q to a line that passes through point P parallel to ector is equal to the length of the component of PQ perpendicular to the line. d PQ sin where is the angle between and ector PQ. 40

41 Since PQ PQ sin, so we hae the shortest distance of Q from L as PQ d Example: Find the distance from the point (0, 0, 0) to the line, Example: x 5 y z 3 5 Find the distance from the point (,, 3) to the line,. x t, y 6t, z 3 4

42 Example: Find the shortest path from the point Q(, 0, - ) to the line l : x 3 y z 6. Planes in Space Suppose that is a plane. Point P x, y, ) ( 0 0 z0 and Q ( x, y, z) lie on it. If N ai bj ck is a non-null ector perpendicular (ortoghonal) to, then N is perpendicular to PQ. P ( x0, y0, z 0) Q ( x, y, z ) 4

43 Thus, PQ N 0 x x0, y y0, z z0 a, b, c 0 a ( x 0 x0) b( y y0) c( z z ) 0 -Conclusion- The equation of a plane can be determined if a point on the plane and a ector orthogonal to the plane are known. -Theorem- (Equation of a Plane) The plane through the point P x, y, ) and with the nonzero normal ector has the equation ( 0 0 z0 N a, b, c 43

44 Point-normal form: a x x by y cz z 0 Standard form: ax by cz d with d ax0 by0 cz0 Example:. Gie an equation for the plane through the point (, 3, 4) and perpendicular to the ector 6,5, 4.. Gie an equation for the plane through the point (4, 5, ) and parallel to the ectors A=,0, and B 0,, 4. 44

45 3. Gie parametric equations for the line through the point (5, -3, ) and perpendicular to the plane 6 y 7z 5 x. 6.. Intersection of Two Planes Intersection of two planes is a line. (L) To obtain the equation of the intersecting line, we need a point on the line L which is gien by soling the equations of the two planes. a ector parallel to the line L which is 45

46 If N N N N a, b, c, then the equation of the line L in symmetrical form is Example: x x0 y y0 z z0 a b c Find the equation of the line passing through P(,3,) and parallel to the line of intersection of the planes x + y - 3z = 4 and x - y + z = 0. Answer: x y 3 z

47 6.. Angle between Two Planes -Properties of Two Planes- An angle between the crossing planes is an angle between their normal ectors. cos N N N N Two planes are parallel if and only if their normal ectors are parallel, N N. Two planes are orthogonal if and only if N N 0. Example: Find the angle between plane 3x 4y 0 and plane y z 5 x. 47

48 6..3 Angle between a Line and a Plane Let be the angle between the normal ector N to a plane and the line L. Then we hae cos N where is ector parallel to L. Furthermore, if the angle between the line L and the plane, then N 48

49 sin sin sin N N cos Example: Find the angle between the plane 3x y z 5 and the line x 3 y z

50 6..4 Shortest Distance from a Point to a Plane (a) From a Point to a Plane -Theorem- The distance D between a point Q x, y, ) and the plane ax by cz d is ( z D N PQ N ax by a b cz c d Where P x, y, ) is any point on the plane. ( 0 0 z0 N D P( x0, y0, z0 ) Q( x, y, z) D 50

51 Example: Find the distance D between the point (, -4, -3) and the plane x 3y 6z. Example: ) Show that the line x y z 3 is parallel to the plane 3 y z x. ) Find the distance from the line to the plane in part (a). (b) Between two parallel planes The distance between two parallel planes ax by cz d and ax by cz d is gien by D a d b d c 5

52 Example: Find the distance between two parallel planes x y z 3 and x 4y 4z 7. -Note- Both formulas can also be used to compute the distance between skewed lines. (c) Between two skewed lines N u L Q d u L P 5

53 Assume L and L are skew lines in space containing the points P and Q and are parallel to ectors u and respectiely. Then the shortest distance between L and L is the perpendicular distance between the two lines and its direction is gien by a ector normal to both lines. So, the distance between the two lines is absolute alue of the scalar projection of PQ on the normal ector. d PQ cos N PQ u PQ N u 53

54 Example: Find the shortest distance from P(, -, ) to the plane 3x 7y + z = 5. Example: Find the shortest distance between the skewed lines. l : x = +t, y = -+ t, z = + 4t l : x = -+4s, y = -3s, z = -+s Example: Find the distance between the lines L : i j 3k t( i k) L : x 0, y t, z 3 t 54

55 Example: Find the distance between the lines L through the points A(, 0, - ) and B(-,, 0) and the line L through the points C(3,, -) and D(4, 5, -). 55

CHAPTER 3 : VECTORS. Definition 3.1 A vector is a quantity that has both magnitude and direction.

CHAPTER 3 : VECTORS. Definition 3.1 A vector is a quantity that has both magnitude and direction. EQT 101-Engineering Mathematics I Teaching Module CHAPTER 3 : VECTORS 3.1 Introduction Definition 3.1 A ector is a quantity that has both magnitude and direction. A ector is often represented by an arrow