Multivariable Calculus Fall 09

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					   Multivariable Calculus Fall 09                                                   09/14
   Quiz 1                           Name(PRINT):

1. Let a = i − j + k and b = i + 2j + 3k, where i, j and k are the standard basis vectors
in R3 .
(a). Find the scaler projection of a onto b.
Solution: a =< 1, −1, 1 >, b =< 1, 2, 3 >, then a · b = 1 − 2 + 3 = 2. The scaler pro-
jection of a onto b is
                              a·b            2         2
                                   =√                =
                               |b|      12 + 22 + 32 14
(b). Find a unit vector perpendicular to both a and b.
Solution: Since a × b gives a vector which is perpendicular to both a and b, and

      a × b =< (−1) ∗ 3 − 1 ∗ 2, 1 ∗ 1 − 1 ∗ 3, 1 ∗ 2 − (−1) ∗ 1 >=< −5, −2, 3 >,

the unit vector perpendicular to a and b is then
                                 a×b      < −5, −2, 3 >
                                        =    √
                                |a × b|         38


2. Find a parametric equation for the line passing through (1, 0, 2) and perpendicular to
the plane 2x − 3y + z = 2.
Solution: Since the line is perpendicular to the plane, the normal vector of the plane
< 2, −3, 1 > is just the direction vector of the line, so the equation of the line is

                       x(t) = 1 + 2t; y(t) = 0 − 3t; z(t) = 2 + t


3. Find an equation for the plane passing through (1, 0, 0),(0, 1, 0) and (0, 0, 1).
                                                                  −→                 →
                                                                                     −
Solution: Let P = (1, 0, 0), Q = (0, 1, 0), R = (0, 0, 1), then PQ = (−1, 1, 0), PR =
(−1, 0, 1), the normal vector of the plane going through P, Q, R is
                     −→ − →
                     PQ × PR = (−1, 1, 0) × (−1, 0, 1) = (1, 1, 1).

Therefore, the equation of the plane is

                    (x − 1) + (y − 0) + (z − 0) = x + y + z − 1 = 0.




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