 
Summary: Practice Problems For Final Math 5B
Chapter 1: Vectors, Matrices, and Applications
Practice Problem:
Write an equation for the plane that contains the points (2, 0, 3), (4, 5, 2), and (0, 3, 4) in the
form ax + by + cz = d.
Solution:
Let v = (4, 5, 2)  (2, 0, 3) = (6, 5, 5) and w = (0, 3, 4)  (2, 0, 3) = (2, 3, 1). Then
v × w = i(5  15)  j(6 + 10) + k(18  10) = (10, 16, 28). We can choose n to be any vector
in the same direction as v × w so let n = (5, 8, 14). Then the plane has the form 5x + 8y + 14z = d.
Substituting the point (2, 0, 3) for (x, y, z) and solving for d gives d = 10 + 0 + (42) = 32. So
the plane has the equation 5x + 8y + 14z = 32 .
Practice Problem:
Find a parametric form for the line passing through the point (1, 2) in the direction (3, 4), which we
will call c1(t). Set c1(t) equal to (x, y) and eliminate t to get the line into y = mx + b form. Now find
a different parametrization c2(t) of the same line such that c2(0) = (2, 2) and c2(2) = (5, 6).
Solution:
c1(t) = (1, 2) + t(3, 4) = (1 + 3t, 2 + 4t). Setting (x, y) = (1 + 3t, 2 + 4t) yields x = 1 + 3t and
y = 2 + 4t. Solving the former equation for t yields t = (x  1)/3. Substituting this into the second
equation then gives us y = 4
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