M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY
Interphase Calculus III Worksheet
Instructor: Samuel S. Watson
06 July 2015
Topics. Matrix multiplication, determinants of matrices, lines and planes in space (Section 12.4 in E&P).
1. Consider the function L : R2 → R2 defined by
L( x, y) = ( x + y, − x + y).
Sketch the images of the lines x = 0, x = 1, y = 0, and y = 1 under L (Hint: find the images of the
corners of the square [0, 1] × [0, 1]). What is the area of the image of the unit square [0, 1] × [0, 1]
under L?
2. Consider L( x, y) = ( ax + by, cx + dy) and K ( x, y) = (ex + f y, gx + hy), where a, b, c, d, e, f , g, h
are all constants. Find ( L ◦ K )( x, y).



3 4 −1
−4 4
3. Evaluate  2 −5 7   1 2 .
4 −3 2
0 0
4. Repeat #1 with the linear transformations represented by the following matrices.
2 0
3 2
0 1
1 0
0 2
0 2
−1 0
0 −1
a b
5. The determinant of
is defined to be ad − bc. Find the determinant of each of the
c d
matrices in the previous exercise.
6. Find the determinant of the following matrix.


3 1 −5
 4 2 6 
−1 4 3
7. Define i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1), and expand the following “determinant” by
minors along the first row.


i
j k
 u1 u2 u3 
v1 v2 v3
8. Describe the line in R3 passing through the points (3, −4, 1) and (2, −1, 4).
9. Find the equation of a plane passing through (3, 0, −7) and perpendicular to the vector (−3, 2, −1).
10. Find the distance between the planes x + y − 2z = 3 and x + y − 2z = 0.
11. Consider the line ` given by the parametric equation ( x (t), y(t), z(t)) = (1 − 2t, 3, t). Find the
shortest distance from ` to a point on the line m which is parallel to ` and which passes through
the point (9, 4, 1).