Three-Dimensional Space; Vectors Exercise Set 11.1 1. (a) (0, 0, 0), (3, 0, 0), (3, 5, 0), (0, 5, 0), (0, 0, 4), (3, 0, 4), (3, 5, 4), (0, 5, 4). (b) (0, 1, 0), (4, 1, 0), (4, 6, 0), (0, 6, 0), (0, 1, −2), (4, 1, −2), (4, 6, −2), (0, 6, −2). 2. Corners: (2, 2, ±2), (2, −2, ±2), (−2, 2, ±2), (−2, −2, ±2). z
(–2, –2, 2)
(2, –2, 2)
(–2, 2, 2)
(2, 2, 2)
(–2, –2, –2) (2, –2, –2)
y
(–2, 2, –2) (2, 2, –2)
x
3. Corners: (4, 2, −2), (4,2,1), (4,1,1), (4, 1, −2), (−6, 1, 1), (−6, 2, 1), (−6, 2, −2), (−6, 1, −2). z
(–6, 2, 1) (–6, 1, –2)
(–6, 2, –2) y
(4, 1, 1)
(4, 2, 1)
(4, 1, –2) x
4. (a) (x2 , y1 , z1 ), (x2 , y2 , z1 ), (x1 , y2 , z1 )(x1 , y1 , z2 ), (x2 , y1 , z2 ), (x1 , y2 , z2 ). (b) The midpoint of the diagonal has coordinates which are the coordinates of the midpoints of the edges. The 1 midpoint of the edge (x1 , y1 , z1 ) and (x2 , y1 , z1 ) is (x1 + x2 ), y1 , z1 ; the midpoint of the edge (x2 , y1 , z1 ) and 2 1 1 (x2 , y2 , z1 ) is x2 , (y1 + y2 ), z1 ; the midpoint of the edge (x2 , y2 , z1 ) and (x2 , y2 , z2 ) is x2 , y2 , (z1 + z2 ) . 2 2 1 1 1 Thus the coordinates of the midpoint of the diagonal are (x1 + x2 ), (y1 + y2 ), (z1 + z2 ) . 2 2 2 5. (a) A single point on that line.
(b) A line in that plane.
(c) A plane in 3−space.
6. (a) R(1, 4, 0) and Q lie on the same vertical line, and so does the side of the triangle which connects them. 519