Question

In a school, the swimming pool is shaped as a cuboid, and the breadth of the pool is represented by line \(\frac{-x+2}{-5} = y = z-3\). Two students A and B are swimming along the length of the pool.

Based on the above information, answer the following questions :

(ii) Write the vector equation of the path of student B swimming parallel to the path of student A, if at a certain instant he passes through the point with coordinates \((2, 4, -1)\).

Hint:

Use the point (2, 4, -1) and the direction ratios 1, 4, -9 of A (taking p = 3) in r = a + lambda b.

Answer 1

Step 1: Get the direction from part (i):
Part (i) gave $p=3$. Then A runs along $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$.
Parallel lines share direction ratios, so B has direction ratios $1,4,-9$.

Step 2: Write the line in Cartesian form first:
B passes through $(2,4,-1)$.
So B is $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z+1}{-9}$.

Step 3: Convert to vector form:
Put each ratio equal to $\lambda$: $x=2+\lambda$, $y=4+4\lambda$, $z=-1-9\lambda$.
Collect these as one vector:
\[ \vec{r}=x\hat{i}+y\hat{j}+z\hat{k}=(2\hat{i}+4\hat{j}-\hat{k})+\lambda(\hat{i}+4\hat{j}-9\hat{k}) \]

Final Answer:
The path of B has the vector equation below, with $\lambda$ a real parameter. \[ \boxed{\vec{r}=(2\hat{i}+4\hat{j}-\hat{k})+\lambda(\hat{i}+4\hat{j}-9\hat{k})} \]