Given the two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$, let $\mathbf{c}$ be the projection of $\mathbf{a}$ onto $\mathbf{b}$ - HSC - SSCE Mathematics Extension 1 - Question 6 - 2023 - Paper 1

To find the projection of one vector onto another, we use the projection formula:

$\text{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}$.

Here we need to find the projection of $10\mathbf{a}$ onto $2\mathbf{b}$:

1. Substitute into the projection formula: $\text{proj}_{2\mathbf{b}}(10\mathbf{a}) = \frac{10\mathbf{a} \cdot (2\mathbf{b})}{(2\mathbf{b}) \cdot (2\mathbf{b})} (2\mathbf{b})$

2. Simplify the numerator: $= \frac{20(\mathbf{a} \cdot \mathbf{b})}{4(\mathbf{b} \cdot \mathbf{b})} (2\mathbf{b})$ $= \frac{5(\mathbf{a} \cdot \mathbf{b})}{(\mathbf{b} \cdot \mathbf{b})} (2\mathbf{b})$

Thus, the projection of $10\mathbf{a}$ onto $2\mathbf{b}$ results in:

$\text{proj}_{2\mathbf{b}}(10\mathbf{a}) = 10\text{proj}_{\mathbf{b}}(\mathbf{a})$

Given that the projection of $\mathbf{a}$ onto $\mathbf{b}$ is $\mathbf{c}$, we have: $10\mathbf{c}$. Therefore, the final answer is:

Answer: 10$c$.