Problems for Appendix A

Vectors and Matrices

A.

A.1    Calculate the magnitudes of each of the vectors shown below

A.1.1

A.1.2

A.1.3

A.2    A vector has magnitude 3, and i and j components of 1 and 2, respectively.Â  Calculate its k component.

A.3    Let {i,j,k} be a Cartesian basis.Â  A vector a has magnitude 4 and subtends angles of 30 degrees and 100 degrees to the i and k directions, respectively.Â  Calculate the components of a in the basis {i,j,k}

A.4    Find the dot products of the vectors listed below

A.4.1                   ,

A.4.2                   ,Â Â

A.4.3                   ,

A.5    Calculate the angle between each pair of vectors listed in Problem A.4. Â i.e. find the angleÂ  Â between a and b in each case

A.6    The vectors a and b shown in the figure have magnitudes , .Â  Calculate .

A.7    Find the cross Â products of the vectors listed below

A.7.1                    ,

A.7.2                    ,Â Â

A.7.3                    ,

A.8    The vectors a and b shown in the figure below have magnitudes , .Â  Calculate .Â  What is the direction of ?

A.9    Let a=5i-6k; b=4i+2kÂ  Let r=5k.Â  Express r as components parallel to a and b, i.e. find two scalars Â and Â such that

A.10  Find the direction cosines of the following vectors

A.10.1

A.10.2

A.11  Let , Â , .

A.11.1                Verify that a, b and c are mutually perpendicular and that

A.11.2                In view of (1), three unit vectors Â parallel to a b and c can form a Cartesian basis.Â  Calculate the components of Â in the {i,j,k} basis.

A.12  Let a and b be two unit vectors.Â  Let Â and Â be two scalars,Â  and let c be a vector such that

A.12.1               Prove that

A.12.2               Let {i,j,k} be a Cartesian basis. Consider the three vectors ,Â  ,Â  .Â  For this set of vectors, calculate values for

A.12.3               By substituting values, show that if a, b, c,Â  Â and Â have the values given in (3.2), then

A.12.4               Â In 30 words or less, explain why the vectors used in (3.2) and (3.3) cannot satisfy Â for any values of Â and . (You may use as many mathematical equations and symbols as you like!).

A.13  Let , Â Â be two vectors, and let Â be a third vector with unknown components

A.13.1               Solve the equation

A.13.2               Find a solution (any solution will do) to the equation

A.13.3               Show that the equation Â has more than one solution (e.g. by finding another one!)

A.13.4               Suppose that the vector Â satisfies

Show that Â must be parallel to

A.13.5               Give all the solutions to the equations

A.14  Suppose that the three vectors Â are used to define a (non-Cartesian) basis.Â  This means that a general vector r is expressed as , where Â denote the components of r in the basis .Â  Let Â and Â denote two vectors, and let .Â  Calculate formulas for the components of Â in , i.e. find formulas in terms of Â and Â for Â such that .

A.15  Here is a nice matrix.

A.15.1                Find

A.15.2                Find Â (Donâ€™t try to use the general expression for the inverse of a matrix Â this matrix can be inverted trivially)

A.15.3                Find the eigenvalues and eigenvectors of . (You can write down one of the eigenvalues and eigenvectors by inspection.Â  The other two can be found using the formulae for a Â matrix)

A.16  Consider a square, symmetric matrix

A.16.1                Find the spectral decomposition of , i.e. find a diagonal matrix Â and an orthogonal matrix Â such that

A.16.2                Hence, calculate

A.17  The exponential ofÂ  a matrix is defined as

A.17.1                Show that the exponential of a diagonal matrix Â is given by

A.17.2                Let Â be a square, symmetric Â matrix, and let Â and Â be a diagonal and orthogonal matrix, respectively, such that .Â  Find an expression for Â in terms of Â and

A.17.3                Calculate the exponential of the matrix given in A.16.

A.18  Find a way to compute the log of a square, symmetric matrix.

A.19  A vector displacement field u(x) has components

where Â are the components of the position vector, and Â are constants. Find expressions for the matrix of components ofÂ  grad(u) and find an expression for the divergence of u.

A.20  The figure shows a three noded, triangular finite element.Â  In the basis , the nodes have coordinates (2,1,3); (4,2,3), (3,3,2).Â  All dimensions are in cm.

A.20.1                Find the area of the element.Â  Use vector algebra to do this Â you do not need to calculate the lengths of the sides of the triangle or any angles between the sides.

A.20.2                Find the angles subtended by the sides of the triangle

A.20.3                Find two unit vectors normal to the plane of the element, expressing your answer as components in the basis

A.20.4                Let Â be a basis with Â parallel to the base of the element, Â normal to the element, and Â chosen so as to ensure that Â are right handed basis vectors. Find the components of , Â and Â in the basis . (To do this, first write down , choose one of the solutions to the preceding problem for , and then form Â by taking the cross product of Â and Â )

A.20.5                Set up the transformation matrix Â that relates vector components in Â to those in

A.20.6                The displacement vector at the center of the element has coordinates (4,3,2) mm in the basis .Â  Find its components in the basis

(c) A.F. Bower, 2008
This site is made freely available for educational purposes.
You may extract parts of the text
for non-commercial purposes provided that the source is cited.