Concept:Using right-angled triangle properties to find the angle between vector components.
Solution:The magnitudes of the vectors (5, 4, and 3) satisfy the Pythagorean theorem:
$$5^2 = 4^2 + 3^2 \implies 25 = 16 + 9$$
This means vectors \(\vec{B}\) and \(\vec{C}\) are perpendicular to each other, and \(\vec{A}\) forms the hypotenuse of a right-angled triangle.
The angle \(\theta\) between \(\vec{A}\) (the hypotenuse) and \(\vec{C}\) (the adjacent side) is given by:
$$\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{C}{A} = \frac{3}{5}$$
$$\theta = \cos^{-1}(3/5)$$
Why other options are incorrect:- B is incorrect because it defines the angle between the hypotenuse \(\vec{A}\) and the vector \(\vec{B}\).
- C and D are mathematically incorrect and do not match the geometric configuration of this right-angled triangle.
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