7281 Test Example Questions Fall 2007
1. Find all values of kfor which the vectors [k2, k, 1] and [1,3,2] are orthogonal.
2. Find the angle between two diagonals of a cube. (There are four diagonal, it
does not matter which pair you choose.)
3. What can you say about the angle between two vectors uand vif
(a) ku+vk=kuk+kvk,
(b) ku+vk2=kuk2+kvk2.
4. Find the area of the triangle with vertices (0,1,3), (2,1,4), (1,3,2).
5. Consider the line though (1,2) in the direction [3,4].
(a) Put this line in normal form.
(b) Find the distance between (0,0) and this line.
6. Consider the plane 2x+ 3y+ 6z= 1.
(a) Put this line in parametric form.
(b) Find the distance between (0,0,0) and this line.
7. Find the intersection of 2x+ 3y+ 6z= 1 and 3x+ 2y+ 5z= 3 in parametric
form (i.e., solve these simultaneous equations).
8. Consider the following matrix
A=
3−6 1
2−4 2
−1 2 1
5−10 −1
(a) Find the rank and nullity of A.
(b) Find a basis for the row space, column space, and null space of A.
9. Find the inverse and determinant of the following matrix
B=
123
357
121
10. find constants A,B,C,D,E, such that
x−1
(x+ 1)(x2+ 1)(x2+ 4) =A
x+ 1 +Bx +C
x2+ 1 +Dx +E
x2+ 4