The angle between vector $\stackrel{\to <!--\; \to \; -->}{\mathrm{Q}}$ and the resultant of $(2\stackrel{\to <!--\; \to \; -->}{\mathrm{Q}}+2\stackrel{\to <!--\; \to \; -->}{\mathrm{P}})$ and $(2\stackrel{\to <!--\; \to \; -->}{\mathrm{Q}}-<!--\; -\; -->2\stackrel{\to <!--\; \to \; -->}{\mathrm{P}})$ is:

The correct answer is 0°.

We need to find the angle between vector $\overrightarrow{\mathrm{Q}}$ and the resultant of the two vectors $(2\overrightarrow{\mathrm{Q}}+2\overrightarrow{\mathrm{P}})$ and $(2\overrightarrow{\mathrm{Q}}-2\overrightarrow{\mathrm{P}})$.

Step 1: Find the resultant of the two given vectors.

The resultant is obtained by adding the two vectors together:

\text{Resultant}=(2\overrightarrow{\mathrm{Q}}+2\overrightarrow{\mathrm{P}})+(2\overrightarrow{\mathrm{Q}}-2\overrightarrow{\mathrm{P}})

Step 2: Simplify the expression.

Group the $\overrightarrow{\mathrm{Q}}$ terms and the $\overrightarrow{\mathrm{P}}$ terms separately:

\text{Resultant}=(2\overrightarrow{\mathrm{Q}}+2\overrightarrow{\mathrm{Q}})+(2\overrightarrow{\mathrm{P}}-2\overrightarrow{\mathrm{P}})

\text{Resultant}=4\overrightarrow{\mathrm{Q}}+\overrightarrow{0}

\text{Resultant}=4\overrightarrow{\mathrm{Q}}

Notice that the $\overrightarrow{\mathrm{P}}$ components cancel each other out completely, leaving only $4\overrightarrow{\mathrm{Q}}$.

Step 3: Find the angle between $\overrightarrow{\mathrm{Q}}$ and $4\overrightarrow{\mathrm{Q}}$.

The resultant is $4\overrightarrow{\mathrm{Q}}$, which is simply a scalar multiple (4 times) of the vector $\overrightarrow{\mathrm{Q}}$. Since multiplying a vector by a positive scalar does not change its direction, both $\overrightarrow{\mathrm{Q}}$ and $4\overrightarrow{\mathrm{Q}}$ point in exactly the same direction.

Therefore, the angle between them is 0°.