We are tasked to determine the angle between vectors $\vec{a}$ and $\vec{b}$ based on the given information:

Given:

1. $|\vec{a}| = \sqrt{3}$,
2. $|\vec{b}| = 2$,
3. $|2\vec{a} - \vec{b}| = 2\sqrt{7}$,
4. Find the angle $(\vec{a}, \vec{b})$.

Solution:

Let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the formula for the magnitude of a vector combination: $|2\vec{a} - \vec{b}|^2 = |2\vec{a}|^2 + |\vec{b}|^2 - 2|2\vec{a}||\vec{b}|\cos\theta.$

Step 1: Calculate each term:

• $|2\vec{a}|^2 = (2|\vec{a}|)^2 = (2\sqrt{3})^2 = 12$,
• $|\vec{b}|^2 = 2^2 = 4$,
• $|2\vec{a} - \vec{b}|^2 = (2\sqrt{7})^2 = 28$.

Step 2: Substitute values into the formula:

$28 = 12 + 4 - 2(2\sqrt{3})(2)\cos\theta.$

Simplify: $28 = 16 - 8\sqrt{3}\cos\theta.$

Rearrange: $8\sqrt{3}\cos\theta = 16 - 28 = -12.$

Solve for $\cos\theta$: $\cos\theta = \frac{-12}{8\sqrt{3}} = -\frac{\sqrt{3}}{2}.$

Step 3: Interpret the result:

$\cos\theta = -\frac{\sqrt{3}}{2}$ corresponds to $\theta = 150^\circ$.

Final Answer:

D. $150^\circ$

Would you like more details or have any further questions?

Here are 5 related questions for practice:

1. How do you calculate the magnitude of a vector combination like $2\vec{a} - \vec{b}$?
2. What are the properties of dot products when finding angles between vectors?
3. Can you prove that $\cos\theta = -\frac{\sqrt{3}}{2}$ leads to $\theta = 150^\circ$?
4. What happens if the vectors $\vec{a}$ and $\vec{b}$ are orthogonal? How would $\cos\theta$ change?
5. What is the geometric significance of $|2\vec{a} - \vec{b}|$ in this problem?

Tip: When solving vector problems, always verify if magnitudes and cosine values are consistent with the problem's context.