100hw1soln

100hw1soln - ||--→ AB || is regarded as the base of Δ,...

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Math 100 Homework 1 fall 2010 1. If ( x, y, z ) belongs to the given surface, x 2 + y 2 + z 2 - y = 0 x 2 + y 2 - y + 1 4 + z 2 = 1 4 x 2 + ( y - 1 2 ) 2 + z 2 = 1 4 the distance from ( x, y, z ) to (0 , 1 2 ) is 1 2 . Therefore, the given surface is the sphere centered at (0 , 1 2 , 0) with radius 1 2 . 2. Choose coordinate axes so that the vertices of the given cube are at (0 , 0 , 0) , (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) , (1 , 1 , 0) , (1 , 0 , 1) , (0 , 1 , 1) , (1 , 1 , 1) . Since (0 , 0 , 0), (1 , 1 , 1) are a pair of opposite vertices, a diagonal of the cube is along (1 , 1 , 1). Since (1 , 0 , 0), (0 , 1 , 1) are a pair of opposite vertices, a diagonal of the cube is along ( - 1 , 1 , 1). Now, (1 , 1 , 1) · ( - 1 , 1 , 1) || (1 , 1 , 1) || || ( - 1 , 1 , 1) || = 1 3 . The angle between these two diagonals is cos - 1 1 3 . 3. No! For instance (1 , 0) · (0 , 1) = (1 , 0) · (0 , - 1). 4. No! For instance (1 , 0 , 0) × ((0 , 1 , 0) × (0 , 1 , 0)) = (0 , 0 , 0) ((1 , 0 , 0) × (0 , 1 , 0)) × (0 , 1 , 0) = ( - 1 , 0 , 0) . 5. Let Δ be the triangle with vertices A , B and P . Then, the area of Δ is half of the area of the parallelogram spanned by -→ AP and --→ AB which is 1 2 || -→ AP × --→ AB || . On the other hand, if
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Unformatted text preview: ||--→ AB || is regarded as the base of Δ, the corresponding altitude would be d . Therefore 1 2 ||-→ AP ×--→ AB || = 1 2 d ||--→ AB || or d = ||-→ AP ×--→ AB || ||--→ AB || . 6. The line we want passes through (0 , , 0) and it is along the same direction as the given line, that is, along (2 , 1 , 0). Therefore, a point ( x, y, z ) lies on the line we want if x = 2 t , y = t , z = 0 7. Suppose that x-x a = y-y b = z-z c . 1 Let t = ( x-x ) /a and we have x = x + at , y = y + bt and z = z + tc . Hence ( x, y, z ) lies on L . Conversely, if ( x, y, z ) lies on L , x = x + at, y = y + bt, z = z + tc for some number t . Therefore t = x-x a = y-y b = z-z c . 2...
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This note was uploaded on 12/25/2011 for the course MATH 101 taught by Professor Ching during the Spring '11 term at HKU.

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100hw1soln - ||--→ AB || is regarded as the base of Δ,...

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