Name ______________________________
Worksheet 11.1
Vector Review (Old Test)
1.
Consider the three points A(1,-1,3), B(-1,0,2), and C(3,1,1)
(a) Find the measure of the angle BAC
(b) Find the area of the triangle ABC
(c) Find the equation of the plane containing ABC
(d) Find the distance of the plane from the point Q located at (3,4,6)
(e) Find the volume of the pyramid QABC
2.
(a) Sketch the image of the triangle after reflection over the y-axis followed by
reflection over the line y   x . Label each vertex clearly.
4
B
A
2
C
-5
5
-2
-4
(b) In words describe the single transformation that would produce the
same result.
(c) Sketch the image of the triangle after rotation through 90 clockwise, followed by
a translation by the vector (-2,3). Label each vertex clearly.
4
B
A
2
C
-5
5
-2
-4
(d) Explain why there cannot exist a single 2x2 matrix that achieves the
transformation described in part (c).
3.
Identify in words the transformations represented by the following matrices. Explain
your reasoning. Also, give the equations of all invariant lines. Make sure to sketch!
 3  4

  4  3
(a) 
 1

(b)  2
 3

 2
 2 6

0
2


(c) 
 3

2 
1 

2 
4.
The picture shows the transformation T(ABC)A’B’C’.
6
B'
4
A'
2
C'
C
B
A
5
(a) Give the matrix of the transformation.
(b) Use determinants to find the area of A’B’C’
(c) Find the eigenvalues of the transformation matrix.
(d) Draw the invariant lines on the picture above.
5.
4
B'
C
C'
2
A'
A
B
-5
5
A''
C''
-2
B''
-4
(a) Find the transformation that takes ABC to A’B’C’ and use it to prove that ABC
is congruent to A’B’C’
(b) Find the transformation that takes ABC to A”B”C” and use it to prove that that
ABC is similar to A”B”C”
6. – Use your calculator on this problem.
(a) Find the matrix of the transformation T1, a reflection over the line y = 2x
(b) Find the matrix of T2, a shear parallel to the y axis (1,0)  (1,3)
(c) Find the matrix representing (T1  T2)-1
(Note that in the transformation T1  T2 , T2 is performed before T1)
7.
(a) Find the equation of the line through (1, -1, 2) which is perpendicular to the plane
3x +2y – z = 6. Recall that this needs to be a parametric equation.
(b) Find the coordinates of the point where this line meets the plane.
(c) Find the distance between the two points
(d) Repeat the above process for the point (x1, y1, z1) and the plane ax + by + cz = d
8.
The distance between two skew lines is the length of their mutual perpendicular. Use a method
similar to the above to find the distance between the lines
 x   1   1
     
 y    0   t  1
 z   2   1
     
 x   2    2
     
 y     1  t  1 
z  3   0 
     