Concept:
Geometric properties of vector addition and subtraction.

Formula:
Let the perpendicular vectors be \(\vec{P}\) and \(\vec{Q}\) with \(|\vec{P}| = |\vec{Q}| = a\) and \(\vec{P} \cdot \vec{Q} = 0\).
$$\vec{S} = \vec{P} + \vec{Q}$$
$$\vec{D} = \vec{P} - \vec{Q}$$

Solution:
Take the scalar product of the sum \(\vec{S}\) and the difference \(\vec{D}\):
$$\vec{S} \cdot \vec{D} = (\vec{P} + \vec{Q}) \cdot (\vec{P} - \vec{Q})$$
$$\vec{S} \cdot \vec{D} = \vec{P} \cdot \vec{P} - \vec{P} \cdot \vec{Q} + \vec{Q} \cdot \vec{P} - \vec{Q} \cdot \vec{Q}$$
Since \(\vec{P} \cdot \vec{Q} = 0\) (perpendicular) and \(\vec{P} \cdot \vec{P} = |\vec{P}|^2 = a^2\):
$$\vec{S} \cdot \vec{D} = a^2 - 0 + 0 - a^2 = 0$$
Since their dot product is zero, the sum vector and difference vector are perpendicular to each other.

Why other options are incorrect:

• A is incorrect because parallel vectors would require a non-zero dot product equal to the product of their magnitudes.
• C and D are incorrect because the dot product is exactly zero, which uniquely corresponds to an angle of \(90^\circ\).