Questions: Figure 3.55 The two displacements A and B add to give a total displacement R having magnitude R and direction θ.

Solution Steps

Given vectors \( \mathbf{A} \) and \( \mathbf{B} \) add to give a resultant vector \( \mathbf{R} \). The magnitudes of \( \mathbf{A} \) and \( \mathbf{B} \) are \( A \) and \( B \) respectively, and the angle between \( \mathbf{A} \) and \( \mathbf{R} \) is \( \theta \).

Since \( \mathbf{A} \) and \( \mathbf{B} \) form a right triangle with \( \mathbf{R} \) as the hypotenuse, we can use the Pythagorean theorem to find \( R \): \[ R = \sqrt{A^2 + B^2} \]

The direction \( \theta \) of the resultant vector \( \mathbf{R} \) can be found using the tangent function: \[ \theta = \tan^{-1}\left(\frac{B}{A}\right) \]

Final Answer