91影视

Geometrical entities with magnitude and direction are known as vectors. A vector is represented as a line with an arrow pointing in the direction of the vector, and the length of the line denotes the vector's magnitude. As a result, vectors are represented by arrows and have two points: a beginning point and a terminal point.

To prove following relation:

$鈭�\left(a.\stackrel{铃�}{r}\right)\left(b.\stackrel{铃�}{r}\right)\sin 蠎d蠎d蠁=\frac{4\mathrm{蟺}}{3}\left(a.b\right)$

It is known that:

$$\begin{gathered} a.\stackrel{铃�}{r}=a_x\sin 蠎\cos 蠁+a_y\sin 蠎\sin 蠁+a_z\cos 蠎 \\ I=鈭�\left(a_x\sin 蠎\cos 蠁+a_y\sin 蠎\sin 蠁+a_z\cos 蠎\right)\left(b_x\sin 蠎\cos 蠁+b_y\sin 蠎\sin 蠁+b_z\cos 蠎\right)\sin 蠎d蠎d蠁{鈭�}_0^{2\mathrm{蟺}}\sin 蠁d蠁 \\ ={鈭�}_0^{2\mathrm{蟺}}\cos 蠁d蠁 \\ ={鈭�}_0^{2\mathrm{蟺}}\sin 蠁\cos 蠁d蠁 \\ =0 \\ I=鈭�\left(a_x{b}_x{\sin }^2蠎{\cos }^2蠁+a_y{b}_y{\sin }^2蠎{\sin }^2蠁+a_z{b}_z{\cos }^2蠎\right)\sin 蠎d蠎d蠁 \end{gathered}$$

But

$$\begin{gathered} ={鈭�}_0^{2\mathrm{蟺}}{\cos }^2蠁d蠁 \\ =\mathrm{蟺}, \end{gathered}$$

$$\begin{gathered} {鈭�}_0^{2\mathrm{蟺}}d蠁=2\mathrm{蟺} \\ \mathrm{I}={鈭�}_0^{\mathrm{蟺}}\left[\mathrm{蟺}\left({\mathrm{a}}_{\mathrm{x}}{\mathrm{b}}_{\mathrm{x}}+{\mathrm{a}}_{\mathrm{y}}{\mathrm{b}}_{\mathrm{y}}\right){\sin }^2\mathrm{蠎}+2{\mathrm{蟺补}}_{\mathrm{z}}{\mathrm{b}}_{\mathrm{z}}{\cos }^2\mathrm{蠎}\right]\mathrm{蝉颈苍蠎诲蠎} \end{gathered}$$

But

$$\begin{gathered} {鈭�}_0^{\mathrm{喂喂}}{\sin }^3蠎d蠎=\frac{4}{3} \\ {鈭�}_0^{\mathrm{喂喂}}{\cos }^2蠎\sin 蠎d蠎=\frac{2}{3} \end{gathered}$$

So,

$$\begin{gathered} I=\mathrm{蟺}\left({\mathrm{a}}_{\mathrm{x}}{\mathrm{b}}_{\mathrm{x}}+{\mathrm{a}}_{\mathrm{y}}{\mathrm{b}}_{\mathrm{y}}\right)\frac{4}{3}{\mathrm{蟺补}}_{\mathrm{z}}{\mathrm{b}}_{\mathrm{z}}\frac{2}{3} \\ =\frac{4}{3}\mathrm{蟺}\left({\mathrm{a}}_{\mathrm{x}}{\mathrm{b}}_{\mathrm{x}}+{\mathrm{a}}_{\mathrm{y}}{\mathrm{b}}_{\mathrm{y}}+{\mathrm{a}}_{\mathrm{z}}{\mathrm{b}}_{\mathrm{z}}\right) \\ =\frac{4}{3}\mathrm{蟺}\left(\mathrm{a}.\mathrm{b}\right) \end{gathered}$$