Homework 2. Chapters 1, 2, 3. Vector addition, dot products and cross products

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1 Homework 2. Chapters 1, 2, 3. Vector addition, dot products and cross products 2.1 Dynamics project description Submit a short (handwritten or typed) paragraph describing your physical system. Include (hand-drawn) schematics showing relevant points, particles, bodies, constraints, etc. Include an interesting question that you would like to answer. 2.2 Sine and cosine review. Trigonometry plays a central role in kinematics, particularly in the formation of rotation matrices. Referring to the figure below, express b1 x, b1 y, b2 x,andb2 y in terms of sin(θ) andcos(θ). b2 x 1 θ b2 y 1 θ b1 y b1 x = b1 y = b2 x = b2 y = b1 x 2.3 Right-handed, orthogonal, unitary basis. Draw a right-handed orthogonal (mutually perpendicular) basis consisting of the unit vectors a x, a y, a z. 2.4 Perpendicular vectors. The vectors v 1 = x a x +2a y +3a z and v 2 =4a x +5a y +6a z are expressed in terms of orthogonal unit vectors a x, a y, a z. Find the value of x so v 1 and v 2 are perpendicular. x = 2.5 Column matrices and vectors. 1 The column matrix 2 is identical to the vector a x +2a y +3a z. True/False Vector concepts: Addition The following vector and column matrix addition produce equivalent results. True/False. Note: a x, a y, a z and b x, b y, b z are sets of orthogonal unit vectors. Explain: a x +2a y +3a z + 4b x +5b y +6b z = = Copyright c by Paul Mitiguy 317 Homework 2

2 2.7 Calculating vector dot products with bases. The figure to the right shows a right-handed (dextral) set of orthogonal unit vectors,, n z. The vectors u, v, w are defined as: u = n z v = x + y + z n z w = n z n z (a) Use the distributive law for dot products to write u v in terms of,, n z, etc. u v = 2x + 2y + 2z n z (b) Use the definition of the dot product to calculate,, etc. = = n z = = = n z = n z = n z = n z n z = (c) In view of your previous two results, calculate u v. u v = (d) As shown in Section , the dot product u v is relatively easy to calculate when,, n z are orthogonal unit vectors. When two arbitrary vectors a and b are expressed in terms of orthogonal unit vectors as shown below, the dot product a b can be calculated as a = a x + a y + a z n z b = b x + b y + b z n z a b = a x b x + a y b y + a z b z In view of this short-cut, calculate u v, u w, and v w. u v =2x +3y +4z u w = v w = (e) MG : Modify and submit the following MotionGenesis file to calculate u w and v w. Note: The following commands show the calculation of u v. % File: CalculateDotProductsWithBasis.al % RigidFrame N Variable x, y, z u> = 2*Nx> + 3*Ny> + 4*Nz> v> = x*nx> + y*ny> + z*nz> w> = 5*Nx> - 6*Ny> + 7*Nz> udotv = Dot( u>, v> ) Save CalculateDotProductsWithBasis.all Quit n z 2.8 Getting started with MotionGenesis. Go to and click on Getting Started. Follow the directions up to Vector Operations. Printoutandsubmit your firstdemo.al file with your homework. Continue through Vector Operations and also submit your vectordemo.al. Copyright c by Paul Mitiguy 318 Homework 2

3 2.9 Definition of a dot product and its use for calculating angles. The figure to the right shows a rectangular parallelepiped (block) of sides 2, 3, and 4. Unit vectors,, n z are directed along the sides of the block as shown. The points A, B, C and D are located at corners of the block. (a) Express r C/A (C s position vector from A) in terms of,, n z. r C/A = (b) Find a numerical value for r C/A r C/A. Next, use equation (2.3) to calculate the magnitude of r C/A (the distance from A to C). r C/A r C/A = r C/A = (c) Using equation (2.1), calculate the unit vector u directed from A to C in terms of,, n z. Next, find the unit vector v directed from A to D in terms of,, n z. u = 3 2 v = 13 (d) Calculate BAC, the angle between line AB and line AC. Next, calculate CAD, the angle between line AC and line AD. BAC = CAD = (e) MG : Modify and submit the following MotionGenesis file to calculate r D/A, v, and CAD. % File: DotProductsToCalculateAngles.al % RigidFrame N Point A, B, C, D B.SetPosition( A, -2*Ny> ) C.SetPosition( B, 3*Nx> ) distancefromatoc = GetMagnitude( C.GetPosition(A) ) u> = C.GetPosition(A) / distancefromatoc anglebacrad = GetAngleBetweenVectors( B.GetPosition(A), C.GetPosition(A) ) anglebacdeg = anglebacrad * ConvertUnits( radians, degrees ) Save DotProductsToCalculateAngles.all Quit Copyright c by Paul Mitiguy 319 Homework 2

4 2.10 Dot products and distance calculations. The figure to the right shows a crane whose cab A supports a boom B that swings a wrecking ball C o. To prevent the wrecking ball from accidently destroying nearby cars, the distance between the nearest car, point N o, and the tip of the boom, point BC, must be controlled. N x A L B θ B B θ C L C C o (a) Express the position vector of BC from N o in terms of x, L B, and the unit vectors,andb x. r BC/No = + (b) Using the distributive property for dot-multiplication of vectors, i.e., (a + b) (c + d) = a c + a d + b c + b d express r BC/No r BC/No in terms of x, L B,and b x. r BC/No r BC/No = (c) Using the definition of the dot-product in equation (2.2), calculate b x. b x = (d) Using your previous two results, rewrite r BC/No r BC/No in terms of x, L B,andθ B. r BC/No r BC/No = (e) Using equation (2.3) to calculate the magnitude of r BC/No, express the distance from N o to BC in terms of x, L B,andθ B, and calculate its value whe=20, L B =10, and θ B =30. r BC/N o = = 29.1 (f) Two colleagues are confused by your use of mixed-bases vectors (i.e., r BC/No = x + L B b x ), and ask you to verify the position vector of B from N o canbeexpressedintheuniform-basis as shown below. Use this uniform-basis expression to verify your previous result for r BC/N o. Note: This uniform-basis approach necessitates the simplifying trigonometric identity sin 2 (θ B )+ cos 2 (θ B )=1. r BC/No = [x + L B cos(θ B )] + L B sin(θ B ) (g) Optional : Calculate the distance from N o to C o in terms of x, L B, L C, θ B,andθ C. r Co/No = Copyright c by Paul Mitiguy 320 Homework 2

5 2.11 Calculating vector cross products with bases. The figure to the right shows a right-handed set of orthogonal unit vectors,, n z.thevectorsu, v, w are defined as: u = n z v = x + y + z n z n w = n z z (a) Use the distributive law for cross products to write u v in terms of,, etc. u v = 2x + 2y + 2z n z (b) Use the definition of the cross product to calculate,, etc. = 0 = n z n z = - = = n z = n z = n z = n z n z = (c) In view of your previous two results, calculate u v. u v = (d) Using the determinant method for calculating the cross product proved in Homework 2.12, calculate u v, u w, and v w. u v = (3z 4 y) + (4x 2 z) + (2y 3 x) n z u w = v w = (e) MG : Modify and submit the following MotionGenesis file to calculate u w and v w. Note: The following commands show the calculation of u v. % File: CalculateCrossProductsWithBasis.al % RigidFrame N Variable x, y, z u> = 2*Nx> + 3*Ny> + 4*Nz> v> = x*nx> + y*ny> + z*nz> w> = 5*Nx> - 6*Ny> + 7*Nz> ucrossv> = Cross( u>, v> ) Save CalculateCrossProductsWithBasis.all Quit n z 2.12 Cross products and determinants. Given right-handed orthogonal unit vectors,, n z and two arbitrary vectors a and b that are expressed in terms of,, n z as shown to the right, prove that calculating a b with the distributive property of the cross product happens to be equal to the determinant of the matrix shown to the right. a = a x + a y + a z n z b = b x + b y + b z n z a b = det n z a x a y a z b x b y b z Copyright c by Paul Mitiguy 321 Homework 2

6 2.13 Optional : Cross product as skew symmetric matrix multiplication. Referring to the previous problem, show that the,, n z coefficients 0 -a z a y of a b happen to be equal to the elements that result from a z 0 -a x the following skew symmetric matrix multiplication. -a y a x 0 After counting the number of computer operations required to multiple the 3 3 matrix by the 3 1 matrix (including multiplication by 0), and comparing the number of operations required to calculate the elements of the simplified answer, it is clear that using a matrix multiplication to calculate a cross product is inefficient True/False Cross products and area calculations. One reason that triangles are important is that complex planar objects can be decomposed into triangles. For example, the polygon B in the figure below can be decomposed into triangles. Knowing the area of two-dimensional objects is helpful in various professions. For example, area measurements are necessary in calculating the acreage and costs associated with building and farming. Knowing the mass properties of a polygon is helpful in determining the motion of two-dimensional objects. B 7 B 5 B 4 r B 1/B 0 = 2.0 b x b y B 8 B 6 r B 2/B 0 = 0.5 b x b y r B 3/B 0 = 3.0 b B x b y 3 r B 4/B 0 = 0.2 b x b y B 2 r B 5/B 0 = -0.5 b B B x b y c B 1 r B 6/B 0 = -1.0 b x b y b y r B 7/B 0 = -2.0 b x b y B 9 bz b r B 8/B 0 = -4.0 b x b y x B 0 r B 9/B 0 = -2.0 b x b y One way to calculate the area of an arbitrary polygon B such as the one shown above is to: Label a vertex B 0 and number the remaining vertices sequentially in a counter-clockwise fashion. Form r B i/b 0, the position vector of vertex B i (i =1, 2,...) from vertex B 0 Calculate A 1, the vector-area of the triangle defined by vertices B 0, B 1,andB 2. Similarly, calculate A 2, A 3,...A 8, the vector-areas of the triangles defined by vertices B 0 B 2 B 3, B 0 B 3 B 4,...B 0 B 8 B 9, respectively. The formula for the vector-area of a triangle is b x b y b z A 1 = 1/2 r B 1/B 0 r B 2/B 0 = 2b z A 2 = 1/2 r B 2/B 0 r B 3/B 0 = A 3 =... = 8.6 b z A 4 =... = A 5 =... = 2.25 b z A 6 =... = 1.5 b z A 7 =... = 9b z A 8 = 1/2 r B 8/B 0 r B 9/B 0 = Calculate A = 8 i=1 A i = The polygon s area is the magnitude of A, i.e., Area = Fill in the previous blanks and determine the polygon s area. Compute cross products with the distributive property (a+b) (c+d) =a c + a d + b c + b d and its definition with the right-hand rule (do not use determinants or look up special formulas in a book). Also, use the fact that b x, b y, b z are orthogonal unit vectors. Copyright c by Paul Mitiguy 322 Homework 2

7 2.15 Scalar triple product with bases. The figure to the right shows a right-handed set of orthogonal unit vectors,, n z.thevectorsu, v, w are defined as: u = n z v = x + y + z n z w = n z n z Calculate u v u, u v w, and u v w. Note: Although the order of operations in u v u is unambiguous, parentheses may clarify your work. u v u = u v w = u v w = 27z 45 x 6 y In view of your last two results, u v w is equal/not equal (circle one) to u v w. It is/is not OK to switch the and in the scalar triple product Optional : Scalar triple products and determinants. Given right-handed orthogonal unit vectors,, n z and three arbitrary vectors a, b, andc that are expressed in terms of,, n z as shown to the right, prove that calculating a (b c) happens to be equal to the determinant of the matrix shown to the right. a = a x + a y + a z n z b = b x + b y + b z n z c = c x + c y + c z n z a x a y a z a b c = b x b y b z c x c y c z 2.17 Constructing unit vectors. Form the unit vector u having the same direction as each vector in the table below. Note: Ensure your answer to the last question agrees with your first two answers, i.e., if c =3 or c = - 3. Vector Unit vector 3-3 Note:,, n z are orthogonal unit vectors n z c n z Copyright c by Paul Mitiguy 323 Homework 2

8 2.18 Locating a microphone (2D). Also see Homework A microphone Q is attached to two pegs B and C by two cables. The point of this practical problem is to determine the distance between Q and point N o knowing the peg locations, cable lengths, and the fact that B, C, Q, andn o all lie in the same plane. Introduce whatever identifiers facilitate your work and try to do the problem first using Euclidean geometry - and then try vectors. Note: There are two mathematical answers to this problem, but one is above the ceiling and requires the cables to be in compression. 8 B 9 15 Q 8 C Quantity Distance from B to C Distance from N o to B Length of cable joining B and Q Length of cable joining C and Q Distance between N o and Q Value 15 m 8m 9m 8m 9.01 m N o 2.19 Locating a microphone (3D). A microphone Q is attached to three pegs A, B, andc by three cables. The point of this practical problem is to determine the distance between Q and point N o knowing the peg locations, cable lengths, and the fact that the walls are perpendicular ( easy problem with the right method). 1 A B N o 13 Q C 8 Quantity Distance from A to B Distance from B to C Distance from N o to B Length of cable joining A and Q Length of cable joining B and Q Length of cable joining C and Q Distance between N o and Q Value 20 m 15 m 8m 15 m 13 m 11 m 13.3 m 2.20 A vector revolution in geometry. The relatively new invention of vectors (Gibbs 1900 AD) has revolutionized Euclidean geometry (Euclid 300 BC). For each geometrical quantity below, circle the vector operation(s) (either the dot-product, cross-product, or both) that is most useful for their calculation. Length: Angle: Area: Volume: 1 Hint: See Section and introduce whatever identifiers facilitate your work. Note: Section shows how to solve nonlinear algebraic equations. This problem can also be solved by-hand. Copyright c by Paul Mitiguy 324 Homework 2