The vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are shown below. Using only vector addition and subtraction, express $\mathbf{b}$ in terms of $\mathbf{a}$, $\mathbf{c}$, and $\mathbf{d}$.
Solution
We seek to write $\mathbf{b}$ as the sum of three vectors related to $\mathbf{a}, \mathbf{d},$ and $\mathbf{c}$.
We try to build a sequence of vectors, placed head to tail, that starts at the tail of $\mathbf{b}$ and ends at the head of $\mathbf{b}$.
We begin by noticing that if the directions of $\mathbf{a}$ and $\mathbf{d}$ were negated, we would have a sequence of three vectors that start at the tail of $\mathbf{b}$ and end at the head of $\mathbf{b}$.