VECTORS IN 2-SPACE AND 3-SPACE GEOMETRIC VECTORS VECTORS OPERATIONS DOT PRODUCT; PROJECTIONS CROSS PRODUCT

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1 VECTORS IN -SPACE AND 3-SPACE GEOMETRIC VECTORS VECTORS OPERATIONS DOT PRODUCT; PROJECTIONS CROSS PRODUCT

2 GEOMETRIC VECTORS Vectors can represented geometrically as directed line segments or arrows in -space and 3-space. We denote ectors in lowercase boldface type ( for instance a,b,u, ). The tail of the arrow is called the initial point of the ector, and the tip of the arrow is called the terminal point B Vectors AB A A : the initial point B : the terminal point August 7, 008 Vectors in -space and 3-space

3 GEOMETRIC VECTORS Definition If and w are any two ectors, then ector +w is represented by the arrow from initial point of to terminal point of w where position the ector w so that its initial point coincides with terminal point of u+w? u w w u u+w August 7, 008 Vectors in -space and 3-space 3

4 GEOMETRIC VECTORS Definition If is a nonzero ector and k is a nonzero scalar, then product k is defined to be the ector whose length is k times the length of and whose direction is the same as that of if k>0 and opposite to that if k<0. - August 7, 008 Vectors in -space and 3-space 4

5 GEOMETRIC VECTORS Definition If and w are any two ectors, then the difference of w from is defined by -w = + (-w) -w -w w August 7, 008 Vectors in -space and 3-space 5

6 VECTORS OPERATIONS Vectors in Coordinate Systems Definition If and w are any two ectors in 3-space, k : scalar = (,, 3 ), w = (w,w,w 3 ) then +w = ( +w, +w, 3 +w 3 ) -w = ( -w, -w, 3 -w 3 ) k = (k,k,k 3 ) Vectors operations in -space are similar with ectors operations in 3-space August 7, 008 Vectors in -space and 3-space 6

7 VECTORS OPERATIONS Properties of Vectors Operations If u, and w are ectors in,3-space, k,m : scalar a. u+ = +u e. (u+)+w = u+(+w) b. u+0 = 0+u f. u+(-u) = 0 c. k(mu) = (km)u g. k(u+) = ku+k d. (k+m) u = ku +mu h. u = u Norm a Vector The length of a ector u is often called the norm of u and denoted by u August 7, 008 Vectors in -space and 3-space 7

8 VECTORS OPERATIONS Let u = (u,u ) then u is gien by formula Let u = (u,u,u 3 ) then u is gien by formula u = u + u u = u + u + u 3 Distance between point (ector) Let A(a,a ) and B(b,b ) are two points (ectors) in -space, then distance between A and B is gien by formula d d ( A, B) AB = ( b a ) + ( b a ) = ( A. B) AB = ( b a ) + ( b a ) + ( b a ) = Let A(a,a,a 3 ) and B(b,b,b 3 ) are two points (ectors) in 3-space, then distance between A and B is 3 3 August 7, 008 Vectors in -space and 3-space 8

9 VECTORS OPERATIONS Example Let u = (,,), u =? Solution u = + + = 9 = 3 Example Determine distance between A(,,) and B(,3,4) Solution d ( A.B) = ( ) + ( 3 ) + ( 4 ) = 4 August 7, 008 Vectors in -space and 3-space 9

10 DOT PRODUCT Definition If u and are ectors in or 3 space and θ is the angle between u and then dot product u. is defined by u. = u cosθ, 0 θ π u u θ θ August 7, 008 Vectors in -space and 3-space 0

11 DOT PRODUCT Let u =(u,u,u 3 ) and =(,, 3 ) are ectors in 3-space,we can derie formula dot product u. when entries of u and are known, then using law of cosines PQ = u + u cosθ,0 θ π Where PQ = - u Then dot product u. is gien by formula (after simplifying) u. = u + u + u 3 If u and ectors in -space, then dot product u. is u. = u + u 3 August 7, 008 Vectors in -space and 3-space

12 DOT PRODUCT Formula u. = u cosθ can be written as cos θ = u. u Formula of dot product can be used to obtain information about the angle (θ) between two ectors θ is acute if and only if u. > 0 θ is obtuse if and only if u. < 0 θ = π/ if and only if u. = 0 August 7, 008 Vectors in -space and 3-space

13 DOT PRODUCT Example Calculate sinθ where θ is angle between a=(,,) and b=(,,0) Solution u = = = 4 + = 5 6 cos θ = u. u = = 3 30 u. = = 3 sin θ = - cos θ sin θ = 30 sin θ = 30 August 7, 008 Vectors in -space and 3-space 3

14 DOT PRODUCT Properties of the Dot Product a. u. =.u b. u. (+w) = u. + u.w c. k (u.) = (ku). = u.(k) d.. > 0 if 0 and. = 0 if = 0 August 7, 008 Vectors in -space and 3-space 4

15 DOT PRODUCT An Orthogonal Projection w u w : Projection of u on w + w = u w : Vector component of u orthogonal to w u w w If u and are ectors in or 3-space and 0, then u. proj u = w = k = w = u w We get k alue by substitute w to u. August 7, 008 Vectors in -space and 3-space 5

16 DOT PRODUCT Example Let a =(,-,3) and b = (4,-,). Find the ector component of a along b and ector component of a orthogonal to b Solution Vector component of a along b is orthogonal projection of a on b a. b =5 b = a. b proj b a = w = b b = 5 ( 4,, ) Vector component of a orthogonal to b is w =a-w = ( 8, 6,33) August 7, 008 Vectors in -space and 3-space 6

17 Definition CROSS PRODUCT Let a =(a,a,a 3 ) and b = (b,b,b 3 ) are ectors in 3-space, then cross product a x b is the ector defined by a xb = i a b a b j k a b 3 3 Where i,j,k are standard unit ector i=(,0,0), j=(0,,0) and k=(0,0,) Relationships Cross Product and Dot Product a. a.(axb) = 0 ( axb ortogonal to a) b. b.(axb) = 0 ( axb ortogonal to b) c. axb = a b (a.b) (Lagrange Identity) August 7, 008 Vectors in -space and 3-space 7

18 CROSS PRODUCT Properties of Cross Product If a,b and c are ectors in 3-space and k : scalar,then a. axb = - (bxa) b. ax(b+c) = (axb) +(axc) c. (a+b)xc = (axc) +(bxc) d. k(axb) = (ka)xb = ax(kb) e. ax0 = 0xa = 0 f. axa = 0 August 7, 008 Vectors in -space and 3-space 8

19 CROSS PRODUCT Geometric Interpretation We can derie formula axb using Lagrange Identity. The formula is axb = a b sin θ What is this? a θ b a sinθ b Area of Parallelogram = a b sin θ Area of Triangle = ½. a b sin θ = ½ axb August 7, 008 Vectors in -space and 3-space 9

20 CROSS PRODUCT Example Find the area of triangle determined by the point A(,,3), B(,,) and C(,0,) Solution Let area of ABC triangle C AB = a = (,0,-) Area of Triangle A B AC = b = (,-,-) = ½ axb i j k a x b = 0 = -i -k = (-,0,-) Area of ABC triangle = ( 8) = axb = 8 August 7, 008 Vectors in -space and 3-space 0

21 EXERCISES. Let a = (k,k,) and b = (k,3,-4). Find k a. If angle between a and b is acute b. If angle between a and b is obtuse c. If angle between a and b is orthogonal. Let a =(,-,3) and b = (4,-,). Find the norm of ector component of a along b? 3. Find a unit ector that is orthogonal to both u and? a. u = (,,-), = (3,,) b. u = (,0,-), = (,,-) August 7, 008 Vectors in -space and 3-space

22 EXERCISES 4. Find the area of triangle haing ertices P,Q,R a. P(0,,), Q(-,,0), R(3,,) b. P(,3,4), Q(-,-,), R(3,,) August 7, 008 Vectors in -space and 3-space