We are given three vectors: $s = -i + 2j$, $t = 3i - j$, and $r = 2i + 5j$. We are asked to find the following: (a) $s + t$ (b) $r - s$ (c) $2t + r$ (d) $3s + 2r$ (e) $\frac{1}{2}(2t - r)$

1. Problem Description

We are given three vectors: $s = -i + 2j$, $t = 3i - j$, and $r = 2i + 5j$. We are asked to find the following:

(a) $s + t$

(b) $r - s$

(c) $2t + r$

(d) $3s + 2r$

(e) $\frac{1}{2}(2t - r)$

2. Solution Steps

(a) $s + t = (-i + 2j) + (3i - j) = (-1+3)i + (2-1)j = 2i + j$

(b) $r - s = (2i + 5j) - (-i + 2j) = (2 - (-1))i + (5 - 2)j = (2+1)i + 3j = 3i + 3j$

(c) $2t + r = 2(3i - j) + (2i + 5j) = (6i - 2j) + (2i + 5j) = (6+2)i + (-2+5)j = 8i + 3j$

(d) $3s + 2r = 3(-i + 2j) + 2(2i + 5j) = (-3i + 6j) + (4i + 10j) = (-3+4)i + (6+10)j = i + 16j$

(e) $\frac{1}{2}(2t - r) = \frac{1}{2}(2(3i - j) - (2i + 5j)) = \frac{1}{2}((6i - 2j) - (2i + 5j)) = \frac{1}{2}((6-2)i + (-2-5)j) = \frac{1}{2}(4i - 7j) = 2i - \frac{7}{2}j$

3. Final Answer

(a) $s + t = 2i + j$

(b) $r - s = 3i + 3j$

(c) $2t + r = 8i + 3j$

(d) $3s + 2r = i + 16j$

(e) $\frac{1}{2}(2t - r) = 2i - \frac{7}{2}j$