MIDDLE EAST TECHNICAL UNIVERSITY
MECHANICAL ENGINEERING DEPARTMENT
ME 210 APPLIED MATHEMATICS FOR MECHANICAL ENGINEERS
SPRING 2008
HOMEWORK 1
Submit only the solutions of parts (c) and (e) of Problem 1, part (a) of Problem 3, and Problem 4 until
17:00 on February 28, 2008, Thursday, to room C-207. Late submissions will not be accepted.
IMPORTANT NOTES
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•
•
•
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1.
Neatness will be graded! In preparing your homework report please refer to the ‘How to Submit a
Successful HW Assignment’ section in the course web site.
Show all your calculation steps and state any conclusions clearly.
You may use software packages such as Mathcad®, MATLAB®, Excel®, etc., only if stated in the
question.
Unless otherwise indicated, all vectors belong to 3-dimensional Euclidean space. Vectors are
designated by either bold lower case letters (e.g., v), as used in the textbook, or by an arrow on top
(e.g. v ). However, in your homework solutions, you have to use arrows for the vectors unless you
present your report as a computer print out.
Scalar product of vectors is shown by • and their cross product by × .
Given the vectors u = i + 2 j , v = 2 j + 5k , and w = 3i − 4 j − 7 k , find
a) v • w (Answer: –43)
b) w (Answer: 74 )
c) the angle between u and v
129 172 301 d) the projection of v on w (Answer: −
i+
j+
k)
74
74
74
e) u × v
2. Show that a ⋅ b = a b cos θ for all vectors a and b where θ is the angle between a and b using the
law of cosines.
3. For all vectors a and b satisfying the following the following conditions
a • b = 0 and ( a + b ) • ( a + b ) = 2 a × b
a) show that a = b ,
b) for c = a × b + a + b , if a = 1 , find the angle between c and the plane on which a and b lie.
(Answer: 35.26°)
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4. Consider a plane Π whose unit normal vector pointing outward from the origin is n = ai + bj + ck and
shortest distance from the origin O(0, 0, 0) is d as shown in the figure. Show that all points P(x, y, z)
on this plane satisfy the equation ax + by + cz – d = 0.
5. Points A(2,0,0), B(2,2,0), C(0,2,0), D(0,2,2), E(0,0,2) and F(2,0,2) are six corners of a cube and points
P, Q and R are midpoints of AB, CD, and EF respectively. Using vector algebra, find the area of the
triangle formed by the points P, Q and R. (Answer: 3 3 2 )
z
R
E
D
F
Q
C
A
P
y
B
x
ThisME210/2008Spring/HW1
study source was downloaded by 100000843412369 from CourseHero.com on 03-12-2022 06:43:58 GMT -06:00
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