Vectors

You are given the vectors X = (1,1,1), y = (2,1,1) and z = (6,2,2). (i) Find the Cartesian equation of the plane Π normal to the vector x containing the point (2,1,1). (ii) Find the parametric equation of the line l through the points (2,1,1) and (6,2,2). (iii) If l' is given parametrically by l' = x + ty (with x and y defined above), find the point of intersection of l' with the plane Π. (iv) Find x x y, y x z and [x, y, z]. Give an explanation in geometric terms of what you have found.

Geometry: Angles

Suppose

Measurement of a parallelepiped brick's diagonal

You have a brick of parallelepiped shape. You need to measure its diagonal (a distance between two opposite vertexes remote to the maximum). You have in your disposition a ruler and a rectangular edge of the plane surface (of the table, for example). What should you do to measure the diagonal if 1. You have also a pencil to make marks on the edge of the table. 2. You don't have a pencil but have another identical brick (of the same shape and dimensions).

Algebra : Puzzle Problems

Please see the attached file for the fully formatted problems. 1) Have you ever seen the written form of the Sanskrit language? If so, you probably are amazed at how different this ancient language from India looks from ours. Some English words, however, are based on Sanskrit. For example, cup comes from the Sanskrit work kupo, which means water well. This puzzle requires several water wells. Suppose you need to measure exactly 1 cup of water. All that you have in your kitchen are two containers. The smaller container holds 3 cups and the larger holds 5 cups. How can you use these two containers to measure exactly 1 cup of water? 2) While flying over farmland, a pilot notices the ... click for more

Tangent Circles

Please see the attached file for the fully formatted problems. Two congruent semicircles lie on the diameter of a third semicircle, each tangent to the other two. A small circle is tangent to all three semicircles. If the area of the shaded region is 45pi, what is the length of the radius of the small circle.