Question

If $|\dot A + \dot B| = |\dot A| = |\dot B|$, then the angle between $\dot A$ and $\dot B$ is

• A. 0°
• B. 60°
• C. 90°
• D. 120°

Answer 1

Answer: (D)

120°

Answer 2

The magnitude of the sum of two vectors $\vec{A}$ and $\vec{B}$ is given by the formula: $|\vec{A} + \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos\theta$ where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$.

Given the condition $|\vec{A} + \vec{B}| = |\vec{A}| = |\vec{B}|$. Let $|\vec{A}| = |\vec{B}| = |\vec{A} + \vec{B}| = k$. Substituting these into the formula: $k^2 = k^2 + k^2 + 2(k)(k)\cos\theta$ $k^2 = 2k^2 + 2k^2\cos\theta$

Assuming $k \neq 0$ (for non-trivial vectors), we can divide the equation by $k^2$: $1 = 2 + 2\cos\theta$ Rearranging the terms to solve for $\cos\theta$: $2\cos\theta = 1 - 2$ $2\cos\theta = -1$ $\cos\theta = -\frac{1}{2}$

The angle $\theta$ whose cosine is $-\frac{1}{2}$ is $120^\circ$.