I had originally solved this problem using Calculus optimization. However, the teacher had a different solution available where he first tried setting $\theta$ to be $90$ degrees in order to cancel out the vertical component of $F_A$. He then tried setting $\theta$ to be $70$ degrees so that the two vectors had an angle of $90$ degrees between each other, which turned out to be the minimum for this question.
I had originally gotten $70$ degrees too, but I had used Calculus to solve this problem.
My question is this: What is the significance of $90$ degrees between the two vectors that makes it the right answer. Why was $\theta$ chosen to be $70$ degrees just like that? I'm unsure of the mathematics behind this.
Sadly I don't have any pictures to illustrate this (yet), but:
"Move" $\vec{F_B}$ in your diagram so that its head is at the target point, 950 units out in the direction of the truck. Now, all the tails of the conceivable $\vec{F_B}$ of a given size (or more formally, the points $T-\vec{F_B}$, where $T$ is the target point) form a circle around the target point, and the ones that represent a valid solution are the ones corresponding to intersections between that circle and the line in the direction $\vec{F_A}$. From here it should be clear that the smallest possible value of $\left|\vec{F_B}\right|$ — that is, the smallest radius of the circle — is when the circle is just tangent to the line. Then the angle between $\vec{F_A}$ and $\vec{F_B}$ is a right angle just because this is the angle between a circle's radius to a given point and its tangent at that point.