Math 200, Sample Test 1 No books, calculators or by hly57249

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									     Math 200, Sample Test 1
No books, calculators or notes allowed. Answer all questions.

1. [10]
Let
                   u = 3i + j + 4k and v = i − 3j + 2k.
                   ¯    ¯ ¯      ¯     ¯ ¯ ¯         ¯
Find
(a) u · v
(b) ¯ | ¯
    |u
     ¯
(c) u × v
    ¯
(d) The ¯unit vector in the direction of u − v
(e) Sketch u − v                         ¯ ¯
           ¯ ¯
Solutions: straightforward definitions.

2. [10]
Find the equations of the following planes:
(a) Through the point (2, 4, −1) with normal vector n = 2i+3j+4k
                                                         0). ¯ ¯ ¯
(b) Through the points (1, 3, 2), (3, −1, 6), and (5, 2, ¯
Solutions: Examples 4 and 5, section 13.5

3. [10]
(a) Find the equation of the sphere, centre (1, −4, 3) and radius 5.
What is the intersection of the sphere and the xz−plane?
(c) Find the equation of the sphere that passes through the point (4, 3, −1)
and has centre (3, 8, 1).
Solutions: 13.1, exercises 11 and 13.

4. [10]
A woman walks due west on the deck of a ship at 3mph.
The ship is moving north at a speed of 22mph.
Find the speed and direction of the woman relative to the surface of
the water.
Solution: 13.2, exercise 31.

5. [10]
Evaluate the following integrals.
                              Z
                          I = x cos(4x2 − 1)dx,
                              Z e
                          J=      x ln(x)dx
                                1
Answers
              Z
                                        1    ¡       ¢
                  x cos(4x2 − 1)dx =      sin 4x2 − 1 + k
                                        8
                      Z e
                                        1 2 1
                          x ln(x)dx =     e +
                      1                 4      4

                                    1
Vector Examples
A man runs East at 10km/hr.
The wind appears to him to blow from the North.
He then doubles his speed and now the wind appears to blow from the NE.
Find the direction and speed of the wind.

Let the runner’s initial velocity be u, the actual wind velocity be v and the
                                     ¯
apparent wind velocities be w1 and w2 .                             ¯
What are we given?


                        u = 10i + 0j , v = Ai + Bj
                        ¯     ¯ ¯ ¯         ¯    ¯
                       w1 = 0i − N j
                             ¯     ¯
                                1
                       w2 = W √ (−i − j).
                                 2 ¯ ¯
What do we need to find?

                              A, B, N and W.
What equations do we have?

        actual wind = runners velocity + apparent wind.

Applying this twice:

                         v = u + w1
                         ¯    ¯
                         v = 2u + w2 .
                         ¯     ¯
Now rewrite in component form

                             A = 10
                             B = −N
                                       W
                             A = 20 − √
                                        2
                                   W
                             B = −√ .
                                     2
Then
                                         W
                   B = −N = −10, A = 10, √ = 10
                                          2
Hence
                                         √ 1
                       v = 10i − 10j = 10 2 √ (i − j)
                       ¯     ¯     ¯         2¯ ¯
and the wind is from the NW at 14.14km/hr



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