hw1 - ECE 3025 A Problem Set#1 Solutions 1 A circle...

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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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