The resultant and dot product of two vectors `vec(a) and vec(b)` is equal to the magnitude of `vec(a)`. Show that when the vector `vec(a)` is doubled , the new resultant is perpendicular to `vec(b)` .

Explanation

To solve the problem, we need to show that when vector \(\vec{A}\) is doubled, the new resultant vector is perpendicular to vector \(\vec{B}\). We start with the given condition that the resultant and dot product of vectors \(\vec{A}\) and \(\vec{B}\) is equal to the magnitude of vector \(\vec{A}\).

### Step 1: Understand the given condition
We are given:
\[ |\vec{A} + \vec{B}| = |\vec{A}| \]
This means that the magnitude of the resultant vector \(\vec{A} + \vec{B}\) is equal to the magnitude of vector \(\vec{A}\).
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