Understanding the Problem: Vectors and Division of a Line Segment

The question involves concepts from vector algebra, specifically dealing with position vectors, internal and external division of a line segment, and the condition for two vectors to be perpendicular.

We are given two points P and Q, whose position vectors with respect to the origin O are \({\rm{\vec p}}\) and \({\rm{\vec q}}\), respectively. This means \(\overrightarrow{{\rm{OP}}} = {\rm{\vec p}}\) and \(\overrightarrow{{\rm{OQ}}} = {\rm{\vec q}}\).

A point R divides the line segment PQ internally in the ratio 2:3. Another point S divides the same line segment PQ externally in the ratio 2:3.

We are told that the position vectors of R and S, relative to the origin, which are \(\overrightarrow{{\rm{OR}}}\) and \(\overrightarrow{{\rm{OS}}}\), are perpendicular to each other. We need to find the relationship between the magnitudes of \({\rm{\vec p}}\) and \({\rm{\vec q}}\).

Applying the Section Formula

The position vector of a point that divides a line segment joining points with position vectors \(\vec{a}\) and \(\vec{b}\) in the ratio m:n can be found using the section formula.

Internal Division

For a point R dividing PQ internally in the ratio m:n, its position vector \(\overrightarrow{{\rm{OR}}}\) is given by:

\(\overrightarrow{{\rm{OR}}} = \frac{n \vec{p} + m \vec{q}}{m+n}\)

In this problem, R divides PQ internally in the ratio 2:3. So, m=2 and n=3. Substituting these values:

\(\overrightarrow{{\rm{OR}}} = \frac{3 \vec{p} + 2 \vec{q}}{2+3} = \frac{3 \vec{p} + 2 \vec{q}}{5}\)

External Division

For a point S dividing PQ externally in the ratio m:n, its position vector \(\overrightarrow{{\rm{OS}}}\) is given by:

\(\overrightarrow{{\rm{OS}}} = \frac{m \vec{q} - n \vec{p}}{m-n}\)

In this problem, S divides PQ externally in the ratio 2:3. So, m=2 and n=3. Substituting these values:

\(\overrightarrow{{\rm{OS}}} = \frac{2 \vec{q} - 3 \vec{p}}{2-3} = \frac{2 \vec{q} - 3 \vec{p}}{-1} = 3 \vec{p} - 2 \vec{q}\)

Using the Perpendicularity Condition

We are given that \(\overrightarrow{{\rm{OR}}}\) and \(\overrightarrow{{\rm{OS}}}\) are perpendicular. Two vectors are perpendicular if and only if their dot product is zero.

So, \(\overrightarrow{{\rm{OR}}} \cdot \overrightarrow{{\rm{OS}}} = 0\).

Substitute the expressions for \(\overrightarrow{{\rm{OR}}}\) and \(\overrightarrow{{\rm{OS}}}\) that we found:

\(\left(\frac{3 \vec{p} + 2 \vec{q}}{5}\right) \cdot (3 \vec{p} - 2 \vec{q}) = 0\)

We can multiply both sides by 5:

\((3 \vec{p} + 2 \vec{q}) \cdot (3 \vec{p} - 2 \vec{q}) = 0\)

Now, perform the dot product similar to algebraic expansion \((a+b)(a-b) = a^2 - b^2\):

\((3 \vec{p}) \cdot (3 \vec{p}) + (3 \vec{p}) \cdot (-2 \vec{q}) + (2 \vec{q}) \cdot (3 \vec{p}) + (2 \vec{q}) \cdot (-2 \vec{q}) = 0\)

\(9 (\vec{p} \cdot \vec{p}) - 6 (\vec{p} \cdot \vec{q}) + 6 (\vec{q} \cdot \vec{p}) - 4 (\vec{q} \cdot \vec{q}) = 0\)

Since the dot product is commutative (\(\vec{p} \cdot \vec{q} = \vec{q} \cdot \vec{p}\)), the middle terms cancel out:

\(9 (\vec{p} \cdot \vec{p}) - 4 (\vec{q} \cdot \vec{q}) = 0\)

Recall that the dot product of a vector with itself is the square of its magnitude, i.e., \(\vec{v} \cdot \vec{v} = |\vec{v}|^2\). The magnitude of \(\vec{p}\) is given as p (denoted as \(|\vec{p}|\)), so \(\vec{p} \cdot \vec{p} = |\vec{p}|^2 = p^2\). Similarly, the magnitude of \(\vec{q}\) is q (denoted as \(|\vec{q}|\)), so \(\vec{q} \cdot \vec{q} = |\vec{q}|^2 = q^2\).

Substitute these into the equation:

\(9 p^2 - 4 q^2 = 0\)

Rearranging the terms, we get:

\(9 p^2 = 4 q^2\)

Final Result

The relationship between p and q derived from the condition that \(\overrightarrow{{\rm{OR}}}\) and \(\overrightarrow{{\rm{OS}}}\) are perpendicular is \(9 p^2 = 4 q^2\). Comparing this with the given options, we find that it matches Option 1.

Revision Table: Key Formulas

Additional Information: Vector Concepts

• Position Vector: A position vector represents the location of a point in space relative to a fixed origin. If O is the origin and P is a point, the vector \(\overrightarrow{OP}\) is the position vector of P.
• Section Formula: This formula is used to find the coordinates or position vector of a point that divides a line segment in a given ratio. It applies to both internal and external division. The ratio m:n means the point divides the segment AB such that the distance from A is proportional to m and the distance from B is proportional to n (with signs considered for external division).
• Dot Product: The dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is a scalar quantity defined as \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta\), where \(\theta\) is the angle between the vectors. If the vectors are perpendicular, \(\theta = 90^\circ\), so \(\cos 90^\circ = 0\), making the dot product zero. Conversely, if the dot product is zero (and the vectors are non-zero), the vectors are perpendicular.
• Magnitude of a Vector: The magnitude of a vector \(\vec{v}\) (denoted as \(|\vec{v}|\) or v) represents its length. In component form, if \(\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}\), then \(|\vec{v}| = \sqrt{a^2 + b^2 + c^2}\). The square of the magnitude is \(|\vec{v}|^2 = a^2 + b^2 + c^2\), which is also equal to \(\vec{v} \cdot \vec{v}\).