If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}-\vec{c}=0$, then angle between $\vec{a}$ and $\vec{b}$ is equal to :

Question

Accepted Answer

To determine the angle between the unit vectors $\vec{a}$ and $\vec{b}$, given the condition $\vec{a} + \vec{b} - \vec{c} = 0$, we can start by rearranging the equation:

$$
\vec{a} + \vec{b} = \vec{c}
$$

Since $\vec{a}$, $\vec{b}$, and $\vec{c}$ are unit vectors, their magnitudes are all equal to 1. Therefore, we can take the magnitude of both sides of the equation:

$$
|\vec{a} + \vec{b}| = |\vec{c}|
$$

Given that $|\vec{c}| = 1$, we have:

$$
|\vec{a} + \vec{b}| = 1
$$

Using the formula for the magnitude of the sum of two vectors, we have:

$$
|\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b}}
$$

Since $\vec{a}$ and $\vec{b}$ are unit vectors, $|\vec{a}| = 1$ and $|\vec{b}| = 1$. Substituting these values, we get:

$$
1 = \sqrt{1^2 + 1^2 + 2 \vec{a} \cdot \vec{b}}
$$

Simplifying, we have:

$$
1 = \sqrt{2 + 2 \vec{a} \cdot \vec{b}}
$$

Squaring both sides, we obtain:

$$
1 = 2 + 2 \vec{a} \cdot \vec{b}
$$

Solving for $\vec{a} \cdot \vec{b}$, we find:

$$
2 \vec{a} \cdot \vec{b} = 1 - 2 = -1
$$

$$
\vec{a} \cdot \vec{b} = -\frac{1}{2}
$$

The dot product $\vec{a} \cdot \vec{b}$ is also equal to $|\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. Since both vectors are unit vectors, this simplifies to:

$$
\cos\theta = -\frac{1}{2}
$$

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

Therefore, the angle between $\vec{a}$ and $\vec{b}$ is $2\pi/3$, which corresponds to Option 4.

**Explanation for Incorrect Options:**

1. **Option 1: $\pi/6$** - This would imply $\cos(\pi/6) = \sqrt{3}/2$, which does not match the calculated $\cos\theta = -1/2$.

2. **Option 2: $\pi/3$** - This would imply $\cos(\pi/3) = 1/2$, which is also incorrect as it does not match $\cos\theta = -1/2$.

3. **Option 3: $\pi/2$** - This would imply $\cos(\pi/2) = 0$, which is incorrect because $\cos\theta = -1/2$.

Thus, Option 4 is the only correct choice, as it aligns with the calculated angle between $\vec{a}$ and $\vec{b}$.