### Challenge Problems

Vector $\overrightarrow{B}$ is 5.0 cm long and vector $\overrightarrow{A}$ is 4.0 cm long. Find the angle between these two vectors when $|\overrightarrow{A}+\overrightarrow{B}|=\phantom{\rule{0.2em}{0ex}}3.0\phantom{\rule{0.2em}{0ex}}\text{cm}$ and $|\overrightarrow{A}-\overrightarrow{B}|=\phantom{\rule{0.2em}{0ex}}3.0\phantom{\rule{0.2em}{0ex}}\text{cm}$.

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

The following figure shows a triangle formed by the three vectors $\overrightarrow{A}$, $\overrightarrow{B}$, and $\overrightarrow{C}$. If vector ${\overrightarrow{C}}^{\prime}$ is drawn between the midpoints of vectors $\overrightarrow{A}$ and $\overrightarrow{B}$, show that ${\overrightarrow{C}}^{\prime}=\overrightarrow{C}\text{/}2$.

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 $\phi $ to become a new coordinate system ${\text{S}}^{\prime}$, as shown in the following figure. A point in a plane has coordinates (*x*, *y*) in S and coordinates $\left({x}^{\prime},{y}^{\prime}\right)$ in ${\text{S}}^{\prime}$.

(a) Show that, during the transformation of rotation, the coordinates in ${\text{S}}^{\prime}$ are expressed in terms of the coordinates in S by the following relations:

(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

(c) Show that the distance between points *P* and *Q* is invariant under rotations of the coordinate system. Here, you have to show that