Sample exam for Math 251-510/511
Based solely on 11.1–11.3
c
2007,
Gregory Berkolaiko
Estimated time: 50 minutes.
1. (10 points) Given three points, A = (3, 0, 2), B = (4, 2, 0) and C =
(5, 1, 4), determine if the triangle ABC is isosceles, right, both or neither.
2. (10 points) Find the scalar and vector projections of b = h4, 2, 0i onto
a = h1, 1, 1i.
3. (5 points) Under what circumstances is compa b = compb a ?
4. (15 points) Find the distance from the point P (2, 0, 3) to the sphere
given by the equation
x2 + y 2 + z 2 − 2x + 4y − 2z + 5 = 0.
(Hint: Find the center and the radius of the sphere to start with.)
5. (20 points) Find the two vectors that are
√
• of length 2 and
• perpendicular to a = (1, −1, 0) and
• form π/3 angle with the vector b = (0, 1, 1).
(Hint: cos(π/3) = 1/2.)
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