第2章ベクトル

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 $|\overrightarrow{A}+\overrightarrow{B}|=3.0\text{cm}$.

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