I assumed all vector bundles are over complex number. Let $V$ be a vector bundle over $X$. Then $P(V)$ denotes the projectivization of fibers of $V-0$ as a projective space bundle over $X$. Denote $K(X)$ as the K group of $X$ or $K_0(X)$. If $V$ is a vector bundle over $X$, there is line bundle $H$ over $P(V)$ associated to $V$ by the following. Since there is a map $P(V)\to X$ projection, let $H=\{([w],v)\in P(V)\times V\vert v\in [w]\}$ be the line bundle over $P(V)$.
Theorem 2.2.1 Let $L$ be a line bundle over $X$. Then $$K(P(L\oplus 1))=\frac{K(X)[H]}{(H-1)(LH-1))},$$ where $K(X)\to K(P(L\oplus 1))$ is induced by $P(L\oplus 1)\to X$ projection map, $1$ is the trivial line bundle and $H$ is the line bundle of $P(L\oplus 1)$ under pullback of $L\oplus 1$ along $P(L\oplus 1)\to X$ as described above.
Question: Is the splitting principle consequence or somehow related to Theorem 2.2.1? I could not see how to deduce the splitting principle from this. The reason I feel it looks like splitting principal is that there is projectivization of line bundle with trivial bundle which happens to be the procedure for splitting principle.
Ref. Atiyah, K-theory Chapter 2, pg 46.