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Unformatted text preview: ECE 3025 A Problem Set #1 Solutions 1. A circle, centered at the origin with a radius of 2 units, lies in the xy plane. Determine the unit vector in rectangular components that lies in the xy plane, is tangent to the circle at ( √ 3 , 1 , 0), and is in the general direction of increasing values of y : A unit vector tangent to this circle in the general increasing y direction is t = a φ . Its x and y components are t x = a φ · a x = − sin φ , and t y = a φ · a y = cos φ . At the point ( √ 3 , 1), φ = 30 ◦ , and so t = − sin 30 ◦ a x + cos 30 ◦ a y = 0 . 5( − a x + √ 3 a y ) . 2. By expressing diagonals as vectors and using the definition of the dot product, find the smaller angle between any two diagonals of a cube, where each diagonal connects diametrically oppo- site corners, and passes through the center of the cube: Assuming a side length, b , two diagonal vectors would be A = b ( a x + a y + a z ) and B = b ( a x − a y + a z ). Now use A · B = | A || B | cos θ , or b 2 (1 − 1+1) = ( √ 3 b )( √ 3 b ) cos θ ⇒ cos θ = 1 / 3 ⇒ θ = 70 . 53 ◦ . This result (in magnitude) is the same for any two diagonal vectors. 3. A sphere of radius a , centered at the origin, rotates about the z axis at angular velocity Ω rad/s. The rotation direction is clockwise when one is looking in the positive z direction....
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This note was uploaded on 10/28/2008 for the course ECE 3025 taught by Professor Citrin during the Spring '08 term at Georgia Tech.
- Spring '08