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 :

(i) If the path of student A is represented by line \(\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-3p}\), find the value of p.

Hint:

Rewrite the breadth line with direction ratios 5, 1, 1. The two paths are perpendicular, so the dot product of their directions is zero.

Answer 1

Step 1: Convert the breadth line to direction numbers:
In a cuboid, the length edge is at right angles to the breadth edge.
The breadth line is $\frac{-x+2}{-5}=y=z-3$.
Write it as $\frac{x-2}{5}=\frac{y}{1}=\frac{z-3}{1}$, so the vector along it is $\vec{b}=5\hat{i}+\hat{j}+\hat{k}$.

Step 2: Write the vector along the path of A:
From the given equation, the vector along A is $\vec{a}=\hat{i}+4\hat{j}-3p\hat{k}$.

Step 3: Set the dot product to zero:
Perpendicular vectors have zero dot product, so $\vec{a}\cdot\vec{b}=0$.
\[ (1)(5)+(4)(1)+(-3p)(1)=0 \]
\[ 9=3p \]
So $p=3$.

Final Answer:
Putting the dot product of the two direction vectors equal to 0 gives $p=3$. \[ \boxed{p=3} \]