Question

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ with $|\vec{a}| = 3$, $|\vec{b}| = 5$ and $|\vec{c}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is

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

Hint:

This problem maps directly to a geometric triangle with side lengths 3, 5, and 7. The angle between $\vec{a}$ and $\vec{b}$ when added as vectors ($\vec{a}+\vec{b}=-\vec{c}$) uses the triangle's internal angle framework. You can think of it using the standard law of cosines from trigonometry directly: $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$. Just keep in mind that since $\vec{a}+\vec{b}+\vec{c}=0$, the angle between the vectors matches the supplementary or interior conditions matching the dot product definition exactly!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a null vector sum equation indicating that three vectors form a closed triangle. Given the specific scalar magnitudes of each vector, we want to isolate the structural angle between vectors $\vec{a}$ and $\vec{b}$.

Step 2: Key Formula or Approach:
1. Isolate the vector $\vec{c}$ on one side of the given equation: $\vec{a} + \vec{b} = -\vec{c}$. 2. Take the self dot product (or square both sides) of the vector expression: $$|\vec{a} + \vec{b}|^2 = |-\vec{c}|^2$$ 3. Expand using the vector identity: $$|\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta = |\vec{c}|^2$$ Where $\theta$ represents the angle between vectors $\vec{a}$ and $\vec{b}$.

Step 3: Detailed Explanation:
Let's substitute the given numerical magnitudes into our expanded formula: $$3^2 + 5^2 + 2(3)(5)\cos\theta = 7^2$$ $$9 + 25 + 30\cos\theta = 49$$ $$34 + 30\cos\theta = 49$$ Isolate the term containing $\cos\theta$: $$30\cos\theta = 49 - 34$$ $$30\cos\theta = 15$$ $$\cos\theta = \frac{15}{30} = \frac{1}{2}$$ Since $\cos\theta = \frac{1}{2}$ within the vector angle domain $[0, \pi]$, we take the inverse cosine: $$\theta = \frac{\pi}{3}\ (\text{or}\ 60^\circ)$$

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