Vector/Matrix operations. *Remember: All parts of HW 1 are due on 1/31 or 2/1

Transcription

1 Lecture 4: Topics: Linear Algebra II Vector/Matrix operations Homework: HW, Part *Remember: All parts of HW are due on / or /

2 Solving Axb Row reduction method can be used Simple operations on equations allowed Goal: Triangular form (ones on diagonal) You can: Swap rows Multiply any row by a scalar number Add/subtract any row from another

3 Example y x y x y x y x () + y x 9 y x () 9) ( 6 8() 9) ( + + Swap rows and Multiply row by / Subtract row from row

4 Overview of D vectors - Coplanar vectors are those which are fully contained within a plane - A plane is formed by two orthogonal axes (the x and y axes) - The components of the vector along the x and y axes are termed the rectangular components - Although we live in a D world, many problems can be effectively analyzed in D sufficient to extract design guidelines - Why analyze in D? Simplicity Components in the rd dimension are small

5 ADDITION OF A SYSTEM OF COPLANAR VECTORS We resolve vectors into components using the x and y axis system. Each component of the vector has a magnitude and a direction. The component directions are based on the x and y axes. We use the unit vectors i and j to designate the x and y axes.

6 For example, F F x i + F y j or F' F' x i + ( F' y ) j The x and y axis are always perpendicular to each other. Together, they can be directed at any inclination.

7 VECTOR RESOLUTION y A x A(cosθ) A A y A y A(sinθ) A (A x )i + (A y )j θ A x x

8 ADDITION OF SEVERAL COPLANAR VECTORS Recognize: Vectors are concurrent Step is to resolve each vector into its components. Step is to add all the x- components together, followed by adding all the y components together. These two totals are the x and y components of the resultant vector. Step is to find the magnitude and angle of the resultant vector.

9 An example of the process: STEP Break the three vectors into components, then add them (Cartesian vector notation) F R F + F + F STEP F x i + F y j F x i + F y j + F x i F y j (F x F x + F x ) i + (F y + F y F y ) j (F Rx ) i + (F Ry ) j

10 Once you know the components, you can then represent the resultant -D vector with a magnitude and angle STEP

11 The direction of a vector can also be defined by the slope, which in turn can be used to solve for vector components: F x F Similar triangles F y b a c F x /F a/c F y /F b/c

12 Example of D vector addition (ON CHALK BOARD)

13 CARTESIAN VECTOR REPRESENTATION OF A D VECTOR Consider a box with sides A X, A Y, and A Z meters long. The vector A can be defined as A (A X i + A Y j + A Z k) m Note the units-this is termed a position vector Same rules apply to all vectors in Cartesian space The magnitude of vector A can be calculated based on the Cartesian components A (A X + A Y + A Z ) ½

14 DIRECTION OF A D CARTESIAN VECTOR DIRECTION COSINES The direction or orientation of vector A is defined by the angles ɑ, β, and γ. These angles are measured between the vector and the positive X, Y and Z axes, respectively. Their range of values are from to 8 Using trigonometry, direction cosines are found using These angles are not independent. They must satisfy the following equation: cos ² α + cos ² β + cos ² γ

15 ADDITION OF -D CARTESIAN VECTORS Once individual vectors are written in Cartesian form, it is easy to add or subtract them. The process is essentially the same as when -D vectors are added. For example, if A A X i + A Y j + A Z k and B B X i + B Y j + B Z k, then A + B (A X + B X ) i + (A Y + B Y ) j + (A Z + B Z ) k or A B (A X - B X ) i + (A Y - B Y ) j + (A Z - B Z ) k.

16 N 45 Specify the magnitude and direction of the net force acting on the femur in this hypothetical loading scenario. 45 N Free-Body Diagram: A simplified sketch that depicts all forces acting on an object of interest 75 N

17 CROSS PRODUCT The cross product of two vectors A and B results in another vector, C, i.e., C A B. The magnitude and direction of the resulting vector can be written as C A B A B sin θ u C As shown, u C is the unit vector perpendicular to both A and B vectors (or to the plane containing the A and B vectors).

18 CROSS PRODUCT (continued) The right-hand rule is a useful tool for determining the direction of the vector resulting from a cross product. For example: i j k

19 CROSS PRODUCT (continued) The cross product can be written as a x determinant. Each component can be determined using determinants.

20 k j i k j i b a k j i b a 4 8 ) (4 ) ( ) (4 so s r q p where then 4, qr ps k j i b a EXAMPLE OF VECTOR CROSS PRODUCTS In MATLAB Calculate the determinant of a x matrix (verify by hand) Calculate the determinant of a x matrix (Ddet(A)) Perform a cross product Ccross(A,B)

21 Matrix division: Not defined A B

22 Matrix Inversion: In arithmetic there are three fundamental concepts that we need to have defined. We need an additive identity, a multiplicative identity and an inverse. Scalar arithmetic we have all three. () Zero is an additive identity. You add zero to any other number and you get that number back. () The number is a multiplicative identity. times any other number gives you the number back. () An inverse is the reciprocal of a number. A number times its inverse is. + q q q q We have the equivalents in matrix arithmetic. q q The zero matrix is the additive identity. The identity matrix is the multiplicative identity. A matrix can have an inverse, A -. *Not all matrices have inverses, just as zero doesn t have a reciprocal.

23 Two rules for matrix inversion. Not all matrices have inverses. (invertible matrices have non-zero determinants). Only square matrices can have inverses. The inverse of a x matrix is easily calculated A a c b d A d c b a A where A is referred to as the determinant of the matrix A and defined for a x matrix as: ad-bc

24 Example A 4 A 4 A 4 4 ( 6) 4 Check (A A - I) In MATLAB Verify direct and indirect matrix inversion: Direct: A - inv(a)

25 General way to A - Instead of setting up Axb for row reduction, use A I (identity matrix) Use row reduction to get I A - Will require multiple steps for elements above diagonal