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

To find the projection of a vector onto another, we can use the formula for the projection of a vector $a$ onto a vector $b$, given by:

ext{proj}_b a = rac{a ullet b}{b ullet b} b

In this case, we want to calculate the projection of $10a$ onto $2b$:

1. First, we can simplify the projection formula by substituting $10a$ for $a$ and $2b$ for $b$:

   ext{proj}_{2b} (10a) = rac{(10a) ullet (2b)}{(2b) ullet (2b)} (2b)

2. Calculating the inner products:

   • The dot product (10a) ullet (2b) = 20 (a ullet b)
   • The dot product (2b) ullet (2b) = 4 (b ullet b)

3. Substituting these results back into the projection formula:

   ext{proj}_{2b} (10a) = rac{20 (a ullet b)}{4 (b ullet b)} (2b)

4. Simplifying further:

   = 5 rac{(a ullet b)}{(b ullet b)} (2b) $= 10 ext{proj}_b a$

5. Given that $c$ is defined as the projection of $a$ onto $b$: $= 10c$

Therefore, the projection of $10a$ onto $2b$ is $10c$.
The answer is C. $10c$.