QExample - 2: HC Verma: Concepts of Physics - Volume 1 Physics and Mathematics Example - 2 Solution: Problem: Two vectors, A and B , each have a magnitude of 5…
When calculating the magnitude of $\vec{A} - \vec{B}$, students often incorrectly substitute the original angle $\theta$ into the formula, failing to recognize that the angle between $\vec{A}$ and $-\vec{B}$ is $180^\circ - \theta$. This oversight leads to using $\cos(60^\circ)$ instead of $\cos(120^\circ)$, resulting in an incorrect magnitude for the difference vector.
Problem:
Two vectors, A and B, each have a magnitude of 5 units. The angle between them is 60 degrees. Find the magnitude of:
(a) The sum of the vectors (A + B) (b) The difference of the vectors (A - B)
Step-by-Step Verified Solution
Solution:
Given:
- |A| = 5 units
- |B| = 5 units
- Angle between A and B (θ) = 60°
(a) Magnitude of the Sum (A + B):
- We use the law of cosines to find the magnitude of the resultant vector (A + B):
- |A + B| = √(|A|² + |B|² + 2|A||B|cos(θ))
- Substitute the given values:
- |A + B| = √(5² + 5² + 2 * 5 * 5 * cos(60°))
- |A + B| = √(25 + 25 + 50 * (1/2))
- |A + B| = √(25 + 25 + 25)
- |A + B| = √75
- |A + B| = 5√3 units
(b) Magnitude of the Difference (A - B):
- We can find the magnitude of (A - B) by considering it as the sum of A and -B. The angle between A and -B is 180° - 60° = 120°.
- We use the law of cosines again:
- |A - B| = √(|A|² + |-B|² + 2|A||-B|cos(120°))
- Substitute the given values (|−B| = |B| = 5):
- |A - B| = √(5² + 5² + 2 * 5 * 5 * cos(120°))
- |A - B| = √(25 + 25 + 50 * (-1/2))
- |A - B| = √(25 + 25 - 25)
- |A - B| = √25
- |A - B| = 5 units
Answers:
(a) The magnitude of the sum of the vectors is 5√3 units.
(b) The magnitude of the difference of the vectors is 5 units.