math226fall07pmt1

math226fall07pmt1 - 1 Given the point P(3 2 4 a what is its...

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1 . 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. b) Write the projections of P on xz-, xy-, and yz-planes. 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 . 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. 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 ; b) the direction angles of ~a (the angles between ~a and the basis vectors ~ i, ~ j, ~ k ; c) the vector and scalar projections of ~a onto ~ b ; 5. Given three points A (1 , 0 , 1) , B ( - 1 , - 1 , 2) , C (1 , 1 , 2) , a) find ~a = A ~ B and ~ b = A ~ C ; find a vector orthogonal to the plane through A, B and C ; b) write a scalar equation of the plane through the points A, B, C ; Find the distance from D (1 , - 1 , 1) to that plane; c) write the vector equation for the line segment connecting A and B ; d) Find the distance from the point D (1 , - 1 , 1) to the line through A and B ; 6. Given a curve ~ r ( t ) = t ~ i + 2 2 3 t 3 / 2 ~ j + 1 2 t 2 ~ k, 0 t 2 , a) find for each t the tangential vector at
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