91影视

Let u, v and w be vectors in ${\mathrm{鈩�}}^3$. Prove:

$u脳(v+w)=u脳v+u脳w\mathrm{and}(u+v)脳w=u脳w+v脳w$

Let $u=(1,0,鈭�8),v=(0,1,6)\mathrm{and}w=(鈭�1,9,3)$

Firstly proving role="math" localid="1649920209467" $u脳(v+w)=u脳v+u脳w$

Now finding the value of $u脳(v+w)$

role="math" localid="1649921394781" $$\begin{gathered} u脳(v+w)=(1,0,-8)脳\left\{(0,1,6)+(-1,9,3)\right\} \\ u脳(v+w)=(1,0,-8)脳(0+(-1),1+9,6+3) \\ u脳(v+w)=(1,0,-8)脳(-1,10,9) \\ u脳(v+w)=\left[\begin{array}{ccc}i& j& k\\ 1& 0& -8\\ -1& 10& 9\end{array}\right] \\ u脳(v+w)=i\left|\begin{array}{cc}0& -8\\ 10& 9\end{array}\right|-j\left|\begin{array}{cc}1& -8\\ -1& 9\end{array}\right|+k\left|\begin{array}{cc}1& 0\\ -1& 10\end{array}\right| \\ u脳(v+w)=i(9-(-80\left)\right)-j(9-8)+k(10-0) \\ u脳(v+w)=89i-j+10k \end{gathered}$$

$$\begin{gathered} u脳v+u脳w=\left[\begin{array}{ccc}i& j& k\\ 1& 0& -8\\ 0& 1& 6\end{array}\right]+\left[\begin{array}{ccc}i& j& k\\ 1& 0& -8\\ -1& 9& 3\end{array}\right] \\ u脳v+u脳w=\left[i\left|\begin{array}{cc}0& -8\\ 1& 6\end{array}\right|-j\left|\begin{array}{cc}1& -8\\ 0& 6\end{array}\right|+k\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|\right]+\left[i\left|\begin{array}{cc}0& -8\\ 9& 3\end{array}\right|-j\left|\begin{array}{cc}1& -8\\ -1& 3\end{array}\right|+k\left|\begin{array}{cc}1& 0\\ -1& 9\end{array}\right|\right] \\ u脳v+u脳w=\left\{i(6-(-8\left)\right)-j(6-0)+k(1-0)\right\}+\left\{i(0-(-72\left)\right)-j(3-8)+k(9-0)\right\} \\ u脳v+u脳w=14i-6j+k+72i+5j+9k \\ u脳v+u脳w=89i-j+10k \end{gathered}$$

Now we prove that$u脳(v+w)=u脳v+u脳w$

Firstly finding the value of $(u+v)脳w$

role="math" localid="1649922089509" $$\begin{gathered} (u+v)脳w=\left\{(1,0,-8)+(0,1,6)\right\}脳(-1,9,3) \\ (u+v)脳w=(1+0,0+1,-8+6)脳(-1,9,3) \\ (u+v)脳w=(1,1,-2)脳(-1,9,3) \\ (u+v)脳w=\left[\begin{array}{ccc}i& j& k\\ 1& 1& -2\\ -1& 9& 3\end{array}\right] \\ (u+v)脳w=i\left|\begin{array}{cc}1& -2\\ 9& 3\end{array}\right|-j\left|\begin{array}{cc}1& -2\\ -1& 3\end{array}\right|+k\left|\begin{array}{cc}1& 1\\ -1& 9\end{array}\right| \\ (u+v)脳w=i(3-(-18\left)\right)-j(3-2)+k(9-(-1\left)\right) \\ (u+v)脳w=21i-j+10k \end{gathered}$$

$$\begin{gathered} u脳w+v脳w=\left[\begin{array}{ccc}i& j& k\\ 1& 0& -8\\ -1& 9& 3\end{array}\right]+\left[\begin{array}{ccc}i& j& k\\ 0& 1& 6\\ -1& 9& 3\end{array}\right] \\ u脳w+v脳w=\left[i\left|\begin{array}{cc}0& -8\\ 9& 3\end{array}\right|-j\left|\begin{array}{cc}1& -8\\ -1& 3\end{array}\right|+k\left|\begin{array}{cc}1& 0\\ -1& 9\end{array}\right|\right]+\left[i\left|\begin{array}{cc}1& 6\\ 9& 3\end{array}\right|-j\left|\begin{array}{cc}0& 6\\ -1& 3\end{array}\right|+k\left|\begin{array}{cc}0& 1\\ -1& 9\end{array}\right|\right] \\ u脳w+v脳w=\left\{i(0-(-72\left)\right)-j(3-8)+k(9-0)\right\}+\left\{i(3-54)-j(0-(-6\left)\right)+k(0-(-1\left)\right)\right\} \\ u脳w+v脳w=72i+5j+9k+51i-6j+k \\ u脳w+v脳w=21i-j+10k \end{gathered}$$

Now we prove that$(u+v)脳w=u脳w+v脳w$