1. **Stating the problem:** Given vectors ū (vertical up), v (left-down), and b (up-right), identify which operation corresponds to the vector shown in the image. 2. **Analyzing vectors:** Vector ū points straight up. Vector b points up-right. The image shows a vector v pointing left-down. 3. **Vector operations:** - $b + ū$: adding ū (up) to b (up-right) would result in a vector pointing more upwards and to the right than b. - $b - ū$: subtracting ū (up) from b (up-right) lowers b vertically, so the resulting vector points up-right but less vertically than b. - $ū - b$: subtracting b (up-right) from ū (up) results in vector ū plus the opposite of b (down-left), moving from pointing straight up to left-down. 4. Since the given vector v points left-down, $ū - b$ matches the direction of v. --- 5. **Stating the problem:** Given vectors a (right), ē (right, shorter), ū (up), v (left), w (down), and d (right, length similar to ē), identify which vector equals $2d$. 6. **Analyzing vectors:** Vector d points right with length similar to ē. Twice d, or $2d$, would be a vector pointing right but twice as long as d. 7. Vector $a$ points right and is longer than ē (and d). Vector $2 ē$ doubles vector ē's length to a vector pointing right even longer. 8. Since d and ē have similar lengths, $2 ē$ is the vector pointing right twice the length of ē, matching $2d$. **Final answers:** - Problem 4: $ū - b$ - Problem 5: $2 ē$