math226fall07pmt1s

math226fall07pmt1s - 1 Given the point P(3 2 4 a what is...

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. Given the point P ( - 3 , 2 , 4) , a) what is its distance to xy-plane? xz-plane? yz-plane? Write the equation of the sphere centered at P that touches yz-plane. Answer. The distance to xy plane is 4; The distance to xz plane is 2; The distance to yz plane is 3; The equation of the sphere ( x + 3) 2 + ( y - 2) 2 + ( z - 4) 2 = 3 2 . b) Write the projections of P on xz-, xy-, and yz-planes. Answer . It is ( - 3 , 0 , 4) on xz plane; ( - 3 , 2 , 0) on xy plane, (0 , 2 , 4) on yz plane. 2. Show that the equation represents a sphere, ﬁnd its radius and the center coordinates: x 2 + y 2 + z 2 - 4 x + 2 y - 6 z = 0 . Answer. Completing squares: x 2 + y 2 + z 2 - 4 x + 2 x - 6 x = x 2 - 4 x + 2 2 + y 2 + 2 y + 1 + z 2 - 6 z + 3 2 - 14 = 0 or ( x - 2) 2 + ( y + 1) 2 + ( z - 3) 2 = 14 . Center at (2,-1,3) and radius 14 . 3. Use vectors to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Answer. If A,B,C are vertices of the triangle, then A ~ B - A ~ C = C ~ B and the vector joining the midpoints is ~a = 1 2 A ~ B - 1 2 A ~ C = 1 2 ± A ~ B - A ~ C ² = 1 2 C ~ B. Therefore ~a is parallel to C ~ B and | ~a | = 1 2 | CB | . 4. Given two vectors ~a = h 1 , 2 , 1 i and ~ b = h- 1 , 1 , 2 i ﬁnd a) the angle between ~a and ~ b ; Answer. For the angle α between them cos α = ~a · ~ b | ~a | | ~ b | = 3 6 = 1 2 , We have ~a · ~ b = - 1+2+2 = 3 , | ~a | = 1 + 4 + 1 = 6 =

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math226fall07pmt1s - 1 Given the point P(3 2 4 a what is...

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