To find the distance between the two boats $A$ and $B$ after $t$ hours, we first need to determine their position vectors at any time $t$. The position vectors of the boats at noon (when $t = 0$) and their velocity vectors are given as follows:

• Initial position vector of boat $A$: $\mathbf{r}_A(0) = 3\mathbf{i} + \mathbf{j}$

• Velocity vector of boat $A$: $\mathbf{v}_A = 10\mathbf{i} + 24\mathbf{j}$

• Initial position vector of boat $B$: $\mathbf{r}_B(0) = \mathbf{i} - 2\mathbf{j}$

• Velocity vector of boat $B$: $\mathbf{v}_B = 24\mathbf{i} + 32\mathbf{j}$

1. Position vectors after $t$ hours:

The position vector of each boat after $t$ hours is given by the initial position vector plus the velocity vector multiplied by $t$:

$\mathbf{r}_A(t) = \mathbf{r}_A(0) + \mathbf{v}_A \cdot t = (3\mathbf{i} + \mathbf{j}) + (10\mathbf{i} + 24\mathbf{j})t = (3 + 10t)\mathbf{i} + (1 + 24t)\mathbf{j}$

$\mathbf{r}_B(t) = \mathbf{r}_B(0) + \mathbf{v}_B \cdot t = (\mathbf{i} - 2\mathbf{j}) + (24\mathbf{i} + 32\mathbf{j})t = (1 + 24t)\mathbf{i} + (-2 + 32t)\mathbf{j}$

2. Distance between the boats after $t$ hours:

The distance between the two boats is the magnitude of the difference between their position vectors:

$\text{Distance} = |\mathbf{r}_A(t) - \mathbf{r}_B(t)|$

First, find the difference in their position vectors:

[ \mathbf{r}_A(t) - \mathbf{r}_B(t) = \left[(3 + 10t)\mathbf{i} + (1 + 24t)\mathbf{j}\right] - \left[(1 + 24t)\mathbf{i} + (-2 + 32t)\mathbf{j}\right] ]

Simplifying this:

[ \mathbf{r}_A(t) - \mathbf{r}_B(t) = [(3 + 10t) - (1 + 24t)]\mathbf{i} + [(1 + 24t) - (-2 + 32t)]\mathbf{j} ]

$= (2 - 14t)\mathbf{i} + (3 - 8t)\mathbf{j}$

Now, calculate the magnitude of this vector:

$\text{Distance} = \sqrt{(2 - 14t)^2 + (3 - 8t)^2}$

Expanding the squares:

= \sqrt{(4 - 56t + 196t^2) + (9 - 48t + 64t^2)}$$ $$= \sqrt{(196t^2 - 56t + 4) + (64t^2 - 48t + 9)}$$ $$= \sqrt{260t^2 - 104t + 13}$$ So, the distance between boats $$A$$ and $$B$$ after $$t$$ hours is: $$\text{Distance} = \sqrt{260t^2 - 104t + 13} \text{ kilometres}$$ Would you like any further details or have any questions? ### Follow-up Questions: 1. How would the distance change if the velocities of the boats were doubled? 2. What would be the distance between the boats at $$t = 2$$ hours? 3. How can you find the time at which the boats are closest to each other? 4. What if the velocity of boat B was in the opposite direction? 5. How would the equation change if the boats started at different positions? ### Tip: When solving vector problems, always remember to carefully handle each component separately before combining them.