The projection of the vector 2ari - 3arj on the vector 3ari + 4arj is

Let $\vec{a}$ be the vector $2\hat{i} - 3\hat{k}$.
Let $\vec{b}$ be the vector $3\hat{i} + 4\hat{j}$.

The dot product $\vec{a} \cdot \vec{b}$ is given by:

The magnitude of $\vec{b}$, denoted as $|\vec{b}|$, is given by:

The projection of vector $\vec{a}$ on vector $\vec{b}$ is given by the formula:

Substitute the values calculated in Step 2 and Step 3:

The calculated projection is $\frac{6}{5}$, which corresponds to option E.

The final answer is $\text{E. } \frac{6}{5}$.