Part 4: Application [10 marks]
21. If $\vec{u}$ and $\vec{v}$ are collinear and the magnitude of $\vec{u}$ is 7 times the magnitude of $\vec{v}$, write an expression that represents the sum of $\vec{u}$ and $\vec{v}$. [2 marks]

Step 1 ：Since \(\vec{u}\) and \(\vec{v}\) are collinear, they are scalar multiples of each other. We are given that the magnitude of \(\vec{u}\) is 7 times the magnitude of \(\vec{v}\). Let's represent \(\vec{u}\) and \(\vec{v}\) in terms of their magnitudes and directions.

Step 2 ：Let \(||\vec{v}||\) be the magnitude of \(\vec{v}\) and \(\hat{v}\) be the unit vector in the direction of \(\vec{v}\). Then, we can write \(\vec{v} = ||\vec{v}||\hat{v}\).

Step 3 ：Since the magnitude of \(\vec{u}\) is 7 times the magnitude of \(\vec{v}\), we can write \(\vec{u} = 7||\vec{v}||\hat{v}\).

Step 4 ：Now, we need to find the sum of \(\vec{u}\) and \(\vec{v}\).

Step 5 ：Let's add \(\vec{u}\) and \(\vec{v}\):

Step 6 ：\(\vec{u} + \vec{v} = 7||\vec{v}||\hat{v} + ||\vec{v}||\hat{v}\)

Step 7 ：\(\vec{u} + \vec{v} = (7 + 1)||\vec{v}||\hat{v}\)

Step 8 ：\(\boxed{\vec{u} + \vec{v} = 8||\vec{v}||\hat{v}}\)