Hello, please see the attached picture and answer the problems shown below. Please show your detailed work and write legibly!! :) Thank you so much and if you have any questions, don't hesitate to ask!!!

1 : Let 11 7E 0 be a vector. The set uJ- = {v : v is orthogonal to u.}. a Show that 0 E uJ' ) b) Show that if v, w 6 til then v + w 6 iii. c) Show that if v E ui then for any A 6 R the vector Av is also in ui. ) d Let u = i. Describe and sketch the set ii. 2 : Let y = mm: + in and y = mgac + b2 be two perpendicular lines. Show that m1m2 = 1 using the following steps. Step 1. Parametrize both lines and write them in the form P + tu, where P is a point on the line and u is a direction vector. Step 2. Since the lines are orthogonal, their direction vectors must be orthog- onal. Use this to complete the proof. 3 :Write the equation of the line a) passing through the points A = (2, 4, 3) and B = (0, 1, 5), b) passing through the point P = (3,2,1) and parallel to line l(t) 2 (4t + 3, 7r, 6t + 1). 4 : Write the equation of the plane a) passing through points P = (2,1,0), Q = (1,1,1) and R = (0,3,5) b) orthogonal to line l(t) 2 (2t + 1, 3t + 2,413) and containing the point P = (3,1,1)