Answer ALL the following questions

Consider a vector space of two dimensions (21,22), a point A = (0,1) and a vector v = (1,1). a) What is the point defined by v considered as a position vector? If you move by one unit of length from point A in the direction of v, what is the new point B you will arrive at? b) What is the position vector of a point P derived by moving from point A along v by a distance s? The result is a parametric representation of a line, where the parameter is s. The line contains A and it is parallel to vector v. c) Find vector u that is perpendicular to vector v above. d) Given point A and vector u, provide a vector equation that every point P on the line must satisfy. Reduce this equation to the form ax + bx2 = 1, i.e. calculate a and b in terms of A and u. X1, X2 are the coordinates of P. Hint: the inner product of vector u and a vector parallel to the line is zero. e) Generalize part (b) to the case of a plane in three dimensions. In this case, point A is a point in three dimensions. v will be replaced by two non-parallel vectors v, w that are both parallel to the plane. Complete this description. f) Find vector u which is perpedicular to both v and w. Hint. g) Generalize part (d) to a plane in three dimensions, i.e. given point A and vector u, provide a vector equation that every point P on the plane must satisfy. Can you reduce this equation to the form ax1 + x2 + cx3 = 1, where X1, 22, 23 are the coordinates of P? Hint: the inner product of vector u with a vector parallel to the plane is zero. h) Consider a plane in three dimensions defined by a point A and a normal vector u. Given a point B not on the plane, find the projection B' of point B onto the plane. Hint: Come up with two properties of the projection B'. One property is based on vector B' A being parallel to the plane. Another property is based on vector B' - B being parallel to vector u