Question

If a + b + c = 0. |a| = 3, |b| = 5, |c| = 7, then the angle between a and b is

• A. \(\frac{π}{6}\)
• B. \(\frac{π}{4}\)
• C. \(\frac{π}{3}\)
• D. \(\frac{π}{2}\)

Answer 1

The Correct Option is C

Given three vectors with magnitudes $|a|=3$, $|b|=5$, $|c|=7$ satisfying $a+b+c=0$, we need to find the angle $\theta$ between vectors $a$ and $b$.

1. Express Vector Relationship:
From $a+b+c=0$, we get $c=-(a+b)$

2. Compute Magnitude Squared:
Taking magnitudes squared: $|c|^2 = |a+b|^2 = |a|^2 + 2(a\cdot b) + |b|^2$
$7^2 = 3^2 + 2(a\cdot b) + 5^2$
$49 = 9 + 2(a\cdot b) + 25$

3. Solve for Dot Product:
$49 = 34 + 2(a\cdot b)$
$2(a\cdot b) = 15$
$a\cdot b = \frac{15}{2}$

4. Relate to Angle:
Using the dot product formula: $a\cdot b = |a||b|\cos\theta$
$\frac{15}{2} = 3\times5\times\cos\theta$
$\cos\theta = \frac{15/2}{15} = \frac{1}{2}$

5. Determine Angle:
$\theta = \arccos\left(\frac{1}{2}\right) = 60^\circ$

Final Answer:
The angle between vectors $a$ and $b$ is $\boxed{60}$ degrees.