Question

1. Let u - (1,1,2), v = (1,2,1), and w (2,1,1) in R. and consider • the parallelogram B = {s(3v) + t-w) 0 <s,t<1, s,te R} spanned/formed by the vectors (3v) and (-w), and • the parallelepiped P = {ru + s(3v) + (-w) 0 <T,,t<1, r, s, t€ R} [10] spanned formed by vectors u. (3v). and (-w) We take the parallelogram B as a base of P. (a) Does the ordered vector triple (v xw, 3v, -w), represent the left hand or the right hand orientation of R$? - Always remember to explain your reasoning! (b) Find the area of B. (c) Find the volume of P. (d) Find all vectors x orthogonal to the base B so that the volume of the parallelepiped [10] 1101 P = {rx + s(3v) + (-w)0 <ri8, <1, r, s, te R} spanned formed by vectors x, (3v) and (-w) is equal to the volume of P.

Answer 1

U = (1,1,2), v = (1, 2, 1), W=(2,1,1) in R² ② (vxw, 30,-) Sime (38)xc-W) = -3 (vxw) has divation opposite to vxw, hence this orientation is not right handes : This is left handed orientation. Area of B = 130x1-w)) = 31vxw/ But wew= det li 2 1 = (1, 1,-3) L2J Irowla Titite all : Aree of B= 3511 square unit volume of P = /[ u, (BU), Eus] = { u. Gu xEw) | = 3/1 + 1 -61 = 12 cube unit. @ An unit vestor y orthogonal to the base B is ya xwe (tan) (vw) x=xy for some sear derrigo} and volume of P' = [x, (v), Ew)] |= 3141 [y, v, w] - 314177 (www.foxw) = 3141. VT = 12 1x1= 1 / 1 » *- tai lux w) )