1.1 Vectors in Two and Three Dimensions 1. Here we just connect the point (0, 0) to the points indicated: y a b c -1 1 2 3 x 0.5 1 1.5 2 2.5 3 2. Although more difficult for students to represent this on paper, the figures should look something like the following. Note that the origin is not at a corner of the frame box but is at the tails of the three vectors. -2 0 2 x -2 0 2 y 0 1 2 3 z b a c In problems 3 and 4, we supply more detail than is necessary to stress to students what properties are being used: 3. (a) (3, 1) + (−1, 7) = (3 + [−1], 1 + 7) = (2, 8). (b) −2(8, 12) = (−2 · 8, −2 · 12) = (−16, −24). (c) (8, 9) + 3(−1, 2) = (8 + 3(−1), 9 + 3(2)) = (5, 15). (d) (1, 1) + 5(2, 6) − 3(10, 2) = (1 + 5 · 2 − 3 · 10, 1 + 5 · 6 − 3 · 2) = (−19, 25). (e) (8, 10) + 3((8, −2) − 2(4, 5)) = (8 + 3(8 − 2 · 4), 10 + 3(−2 − 2 · 5)) = (8, −26). 4. (a) (2, 1, 2) + (−3, 9, 7) = (2 − 3, 1 + 9, 2 + 7) = (−1, 10, 9). ( ) ( ) ( ) (b) 1 2 (8, 4, 1) + 2 5, −7, 1 4 = 4, 2, 1 2 + 10, −14, 1 2 = (14, −12, 1). ( ( )) (c) −2 (2, 0, 1) − 6 1 2 , −4, 1 = −2((2, 0, 1) − (3, −24, 6)) = −2(−1, 24, −5) = (2, −48, 10). 5. We start with the two vectors a and b. We can complete the parallelogram as in the figure on the left. The vector from the origin to this new vertex is the vector a + b. In the figure on the right we have translated vector b so that its tail is the head of vector a. The sum a + b is the directed third side of this triangle.