91Ó°ÊÓ

Vector \(\overrightarrow{\mathbf{B}}\) is \(5.0 \mathrm{cm}\) long and vector \(\overrightarrow{\mathbf{A}}\) is 4.0 cm long. Find the angle between these two vectors when \(|\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}|=3.0 \mathrm{cm}\) and \(|\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}|=3.0 \mathrm{cm}\)

What is the component of the force vector \(\overrightarrow{\mathbf{G}}=(3.0 \hat{\mathbf{i}}+4.0 \hat{\mathbf{j}}+10.0 \hat{\mathbf{k}}) \mathrm{N}\) along the force vector \(\overrightarrow{\mathbf{H}}=(1.0 \hat{\mathbf{i}}+4.0 \hat{\mathbf{j}}) \mathrm{N} ?\)

Distances between points in a plane do not change when a coordinate system is rotated. In other words, the magnitude of a vector is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle \(\varphi\) to become a new coordinate system \(\mathrm{S}^{\prime},\) as shown in the following figure. \(\mathrm{A}\) point in a plane has coordinates \((x, y)\) in \(S\) and coordinates \(\left(x^{\prime}, y^{\prime}\right)\) in \(\mathrm{S}^{\prime}\) (a) Show that, during the transformation of rotation, the coordinates in \(S^{\prime}\) are expressed in terms of the coordinates in S by the following relations: \(\left\\{\begin{array}{l}x^{\prime}=x \cos \varphi+y \sin \varphi \\\ y^{\prime}=-x \sin \varphi+y \cos \varphi\end{array}\right.\) (b) Show that the distance of point \(P\) to the origin is invariant under rotations of the coordinate system. Here, you have to show that \(\sqrt{x^{2}+y^{2}}=\sqrt{x^{2}+y^{2}}\) (c) Show that the distance between points \(P\) and \(Q\) is invariant under rotations of the coordinate system. Here, you have to show that $$\sqrt{\left(x_{P}-x_{Q}\right)^{2}+\left(y_{P}-y_{Q}\right)^{2}}=\sqrt{\left(x_{P}^{\prime}-x^{\prime} \rho\right)^{2}+\left(y_{P}^{\prime}-y^{\prime}_{Q}\right)^{2}}$$