Question

$\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ and $|\vec{a}| = 3, |\vec{b}| = 5, |\vec{c}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is

• A. $\frac{\pi}{4}$
• B. $\frac{\pi}{2}$
• C. $\frac{\pi}{3}$
• D. $\frac{\pi}{6}$

Hint:

To find the angle between any two vectors from a zero-sum equation, always isolate those two vectors on one side and square the equation. The resulting dot product term ($2ab\cos\theta$) immediately unlocks the angle.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the sum of three vectors equals zero and their individual magnitudes. We must find the specific angle between vectors $\vec{a}$ and $\vec{b}$.

Step 2: Key Formula or Approach:
To find the angle $\theta$ between $\vec{a}$ and $\vec{b}$, we isolate them on one side of the equation and square both sides. The expansion involves the dot product:
$$|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta$$

Step 3: Detailed Explanation:
Start with the given vector sum:
$$\vec{a} + \vec{b} + \vec{c} = \vec{0}$$ Since we need the angle between $\vec{a}$ and $\vec{b}$, keep them together and move $\vec{c}$ to the other side:
$$\vec{a} + \vec{b} = -\vec{c}$$ Square both sides (which is equivalent to taking the dot product of each side with itself):
$$|\vec{a} + \vec{b}|^2 = |-\vec{c}|^2$$ Expand the left side:
$$|\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b}) = |\vec{c}|^2$$ Substitute $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$:
$$|\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta = |\vec{c}|^2$$ Now, plug in the given magnitudes ($|\vec{a}| = 3, |\vec{b}| = 5, |\vec{c}| = 7$):
$$(3)^2 + (5)^2 + 2(3)(5)\cos\theta = (7)^2$$ $$9 + 25 + 30\cos\theta = 49$$ $$34 + 30\cos\theta = 49$$ Subtract 34 from both sides:
$$30\cos\theta = 15$$ $$\cos\theta = \frac{15}{30} = \frac{1}{2}$$ The angle whose cosine is $\frac{1}{2}$ is $60^\circ$ or $\frac{\pi}{3}$ radians.
$$\theta = \frac{\pi}{3}$$

Step 4: Final Answer:
The angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$, matching option (C).