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.  Calculate its k component.

 

 

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

A.4.3                   ,  

 

 

 

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

 

 

 

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

 

 

 

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

A.7.1                    ,  

A.7.2                    ,    

A.7.3                    ,  

 

 

 

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

 

 

 

 

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

 

 

 

A.10  Find the direction cosines of the following vectors

A.10.1                 

A.10.2                 

 

 

 

A.11  Let ,  , .

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

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

 

 

 

A.12  Let a and b be two unit vectors.  Let  and  be two scalars,  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 .  For this set of vectors, calculate values for

 

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

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

 

 

 

A.13  Let ,   be two vectors, and let  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  has more than one solution (e.g. by finding another one!)

A.13.4               Suppose that the vector  satisfies

 

Show that  must be parallel to  

A.13.5               Give all the solutions to the equations

 

 

 

 

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

 

 

 

A.15  Here is a nice matrix.

 

A.15.1                Find  

A.15.2                Find  (Don’t try to use the general expression for the inverse of a matrix  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.  The other two can be found using the formulae for a  matrix)

 

 

 

A.16  Consider a square, symmetric matrix

 

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

A.16.2                Hence, calculate  

 

 

 

 

A.17  The exponential of  a matrix is defined as

 

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

 

A.17.2                Let  be a square, symmetric  matrix, and let  and  be a diagonal and orthogonal matrix, respectively, such that .  Find an expression for  in terms of  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  are the components of the position vector, and  are constants. Find expressions for the matrix of components of  grad(u) and find an expression for the divergence of u.

 

 

 

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

A.20.1                Find the area of the element.  Use vector algebra to do this  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  be a basis with  parallel to the base of the element,  normal to the element, and  chosen so as to ensure that  are right handed basis vectors. Find the components of ,  and  in the basis . (To do this, first write down , choose one of the solutions to the preceding problem for , and then form  by taking the cross product of  and  )

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

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

 

 

 

 

 

(c) A.F. Bower, 2008
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