If $|\vec{a}|=|\vec{b}|=|\vec{c}|$ and $\vec{a}+\vec{b}=\vec{c}$, then the angle between $\vec{a}$ and $\vec{b}$ is:

Q: If $|\vec{a}|=|\vec{b}|=|\vec{c}|$ and $\vec{a}+\vec{b}=\vec{c}$, then the angle between $\vec{a}$ and $\vec{b}$ is:

To determine the angle between vectors $\vec{a}$ and $\vec{b}$, given that $|\vec{a}| = |\vec{b}| = |\vec{c}|$ and $\vec{a} + \vec{b} = \vec{c}$, we can use vector properties and the cosine rule for vectors. ### Explanation for Option 3: 1. **Given Conditions:** - $|\vec{a}| = |\vec{b}| = |\vec{c}|$ - $\vec{a} + \vec{b} = \vec{c}$ 2. **Using the Magnitude Condition:** - Since $|\vec{a}| = |\vec{b}| = |\vec{c}|$, let $|\vec{a}| = |\vec{b}| = |\vec{c}| = k$. 3. **Vector Addition:** - $\vec{a} + \vec{b} = \vec{c}$ 4. **Magnitude of $\vec{c}$:** - $|\vec{c}|^2 = |\vec{a} + \vec{b}|^2$ - $|\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta$ - Since $|\vec{a}| = |\vec{b}| = |\vec{c}| = k$, we have: \[ k^2 = k^2 + k^2 + 2k^2\cos\theta \] - Simplifying gives: \[ k^2 = 2k^2 + 2k^2\cos\theta \] \[ 0 = k^2 + 2k^2\cos\theta \] \[ 0 = 1 + 2\cos\theta \] \[ \cos\theta = -\frac{1}{2} \] 5. **Angle Calculation:** - The angle $\theta$ for which $\cos\theta = -\frac{1}{2}$ is $\theta = 2\pi/3$. Thus, the angle between $\vec{a}$ and $\vec{b}$ is $2\pi/3$, which corresponds to Option 3. ### Explanation for Why Other Options are Incorrect: 1. **Option 1: $\pi/2$** - If the angle were $\pi/2$, $\cos\theta$ would be 0. This would not satisfy the equation $0 = 1 + 2\cos\theta$. 2. **Option 2: $\pi$** - If the angle were $\pi$, $\cos\theta$ would be -1. This would make the equation $0 = 1 + 2(-1)$, which simplifies to $-1 = 0$, a contradiction. 3. **Option 4: $\pi/4$** - If the angle were $\pi/4$, $\cos\theta$ would be $\sqrt{2}/2$. This would not satisfy the equation $0 = 1 + 2\cos\theta$. Therefore, the only angle that satisfies the given conditions is $2\pi/3$, making Option 3 the correct answer.

Question

Accepted Answer

To determine the angle between vectors $\vec{a}$ and $\vec{b}$, given that $|\vec{a}| = |\vec{b}| = |\vec{c}|$ and $\vec{a} + \vec{b} = \vec{c}$, we can use vector properties and the cosine rule for vectors.

### Explanation for Option 3:

1. **Given Conditions:**
   - $|\vec{a}| = |\vec{b}| = |\vec{c}|$
   - $\vec{a} + \vec{b} = \vec{c}$

2. **Using the Magnitude Condition:**
   - Since $|\vec{a}| = |\vec{b}| = |\vec{c}|$, let $|\vec{a}| = |\vec{b}| = |\vec{c}| = k$.

3. **Vector Addition:**
   - $\vec{a} + \vec{b} = \vec{c}$

4. **Magnitude of $\vec{c}$:**
   - $|\vec{c}|^2 = |\vec{a} + \vec{b}|^2$
   - $|\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta$
   - Since $|\vec{a}| = |\vec{b}| = |\vec{c}| = k$, we have:
     $$
     k^2 = k^2 + k^2 + 2k^2\cos\theta
     $$
   - Simplifying gives:
     $$
     k^2 = 2k^2 + 2k^2\cos\theta
     $$
     $$
     0 = k^2 + 2k^2\cos\theta
     $$
     $$
     0 = 1 + 2\cos\theta
     $$
     $$
     \cos\theta = -\frac{1}{2}
     $$

5. **Angle Calculation:**
   - The angle $\theta$ for which $\cos\theta = -\frac{1}{2}$ is $\theta = 2\pi/3$.

Thus, the angle between $\vec{a}$ and $\vec{b}$ is $2\pi/3$, which corresponds to Option 3.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $\pi/2$**
   - If the angle were $\pi/2$, $\cos\theta$ would be 0. This would not satisfy the equation $0 = 1 + 2\cos\theta$.

2. **Option 2: $\pi$**
   - If the angle were $\pi$, $\cos\theta$ would be -1. This would make the equation $0 = 1 + 2(-1)$, which simplifies to $-1 = 0$, a contradiction.

3. **Option 4: $\pi/4$**
   - If the angle were $\pi/4$, $\cos\theta$ would be $\sqrt{2}/2$. This would not satisfy the equation $0 = 1 + 2\cos\theta$.

Therefore, the only angle that satisfies the given conditions is $2\pi/3$, making Option 3 the correct answer.