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Step-by-Step Solution
Step 1: Understand the Given Condition
We are given that
$|\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}|$.
This situation occurs when two vectors add up in the most straightforward way, meaning one does not diminish the magnitude of the other in any direction.
Step 2: Recall the Formula for the Magnitude of the Sum of Two Vectors
The magnitude of the sum of two vectors $\vec{A}$ and $\vec{B}$ is given by:
$|\vec{A} + \vec{B}|^{2} = |\vec{A}|^{2} + |\vec{B}|^{2} + 2\,|\vec{A}|\,|\vec{B}|\cos\theta,$
where $\theta$ is the angle between the two vectors $\vec{A}$ and $\vec{B}$.
Step 3: Apply the Given Condition
From the problem statement, we also know:
$|\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}|.$
Squaring both sides gives:
$(|\vec{A} + \vec{B}|)^{2} = (|\vec{A}| + |\vec{B}|)^{2}.$
Substitute from the formula in Step 2:
$|\vec{A}|^{2} + |\vec{B}|^{2} + 2\,|\vec{A}|\,|\vec{B}|\cos\theta
= |\vec{A}|^{2} + |\vec{B}|^{2} + 2\,|\vec{A}|\,|\vec{B}|.$
Step 4: Solve for the Angle
Canceling out common terms on both sides, we get:
$2\,|\vec{A}|\,|\vec{B}|\cos\theta = 2\,|\vec{A}|\,|\vec{B}|.$
Dividing both sides by $2\,|\vec{A}|\,|\vec{B}|$ (assuming neither vector has zero magnitude), we get:
$\cos\theta = 1.$
The angle $\theta$ which satisfies $\cos\theta = 1$ is $0^\circ.$
Step 5: Conclude the Result
Hence, for
$|\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}|,$
the vectors $\vec{A}$ and $\vec{B}$ must be pointing in the same direction, i.e., the angle between them is $0^\circ.$
