91影视

The non-zero vectors $\mathbf{u}$and $\mathbf{v}$.

The non-zero vectors $\mathbf{u}=鉄�1,2,0鉄�$and $\mathbf{v}=鉄�2,4,0鉄�$.

The cross-product $\mathbf{u}脳\mathbf{v}$gives:

role="math" localid="1650771821570" $$\begin{gathered} \mathbf{u}脳\mathbf{v}=det\left[\begin{array}{l}\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 2& 4& 0\\ 1& 2& 0\end{array}\end{array}\right] \\ =0\mathbf{i}-0\mathbf{j}-0\mathbf{k} \end{gathered}$$

The cross-product $\mathbf{v}脳\mathbf{u}$gives:

role="math" localid="1650771829212" $$\begin{gathered} \mathbf{v}脳\mathbf{u}=det\left[\begin{array}{l}\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 2& 4& 0\\ 1& 2& 0\end{array}\end{array}\right] \\ =0\mathbf{i}-0\mathbf{j}-0\mathbf{k} \end{gathered}$$

Therefore, the cross-products $\mathbf{u}脳\mathbf{v}$and $\mathbf{v}脳\mathbf{u}$are equal.

Thus, for the non-zero vectors $\mathbf{u}=鉄�1,2,0鉄�$and $\mathbf{v}=鉄�2,4,0鉄�$the equation $\mathbf{u}脳\mathbf{v}=\mathbf{v}脳\mathbf{u}$ holds.

The non-zero vectors $\mathbf{u},\mathbf{v}$and $\mathbf{w}$.

The vectors $\mathbf{u}=鉄�2,1,-3鉄�,\mathbf{v}=鉄�4,0,1鉄�$and $\mathbf{w}=鉄�-2,6,5鉄�$.

The vectors $\mathbf{u}=鉄�2,1,-3鉄�$, and $\mathbf{v}=鉄�4,0,1鉄�$.

$\mathbf{u}脳\mathbf{v}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 2& 1& -3\\ 4& 0& 1\end{array}\right|$

Now, calculate, $\mathbf{u}脳\mathbf{v}$.

$$\begin{gathered} \mathbf{u}脳\mathbf{v}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 2& 1& -3\\ 4& 0& 1\end{array}\right| \\ =\mathbf{i}(1-0)-\mathbf{j}(2+12)+\mathbf{k}(0-4) \\ =\mathbf{i}-14\mathbf{j}-4\mathbf{k} \end{gathered}$$

Hence, the value of $\mathbf{u}脳\mathbf{v}$is $\mathbf{i}-14\mathbf{j}-4\mathbf{k}$

Now, calculate $(\mathbf{u}脳\mathbf{v})脳\mathbf{w}$

$$\begin{gathered} (\mathbf{u}脳\mathbf{v})脳\mathbf{w}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 1& -14& -4\\ -2& 6& 5\end{array}\right| \\ =\mathbf{i}(-70+24)-\mathbf{j}(5-8)+\mathbf{k}(6-28) \\ =-46\mathbf{i}+3\mathbf{j}-22\mathbf{k} \end{gathered}$$

Hence, the value of $(\mathbf{u}脳\mathbf{v})脳\mathbf{w}$is $-46\mathbf{i}+3\mathbf{j}-22\mathbf{k}$.

Consider vectors $\mathbf{v}=鉄�4,0,1鉄�$and $\mathbf{w}=鉄�-2,6,5鉄�$

$\mathbf{v}脳\mathbf{w}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 4& 0& 1\\ -2& 6& 5\end{array}\right|$

Now, calculate, $\mathbf{v}脳\mathbf{w}$.

$$\begin{gathered} \mathbf{v}脳\mathbf{w}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 4& 0& 1\\ -2& 6& 5\end{array}\right| \\ =-6\mathbf{i}-22\mathbf{j}+24\mathbf{k} \end{gathered}$$

Hence, the value of$\mathbf{v}脳\mathbf{w}$is $-6\mathbf{i}-22\mathbf{j}+24\mathbf{k}$.

Now, calculate $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})$

$$\begin{gathered} \mathbf{u}脳(\mathbf{v}脳\mathbf{w})=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 2& 1& -3\\ -6& -22& 24\end{array}\right| \\ =\mathbf{i}(24-66)-\mathbf{j}(48-18)+\mathbf{k}(-44+6) \\ =-42\mathbf{i}-30\mathbf{j}-38\mathbf{k} \end{gathered}$$

Hence, the value of $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})$is $-42\mathbf{i}-30\mathbf{j}-38\mathbf{k}$.

Therefore, for the vectors$\mathbf{u}=鉄�2,1,-3鉄�,\mathbf{v}=鉄�4,0,1鉄�$and $\mathbf{w}=鉄�-2,6,5鉄�$the result $(\mathbf{u}脳\mathbf{v})脳\mathbf{w}鈮�\mathbf{u}脳(\mathbf{v}脳\mathbf{w})$ holds.

The vectors $\mathbf{u},\mathbf{v}$and $\mathbf{w}$.

Consider the vectors $\mathbf{u}=鉄�1,2,0鉄�,\mathbf{v}=鉄�2,4,0鉄�$and $\mathbf{w}=鉄�4,8,0鉄�$.

Consider the vectors $\mathbf{u}=鉄�1,2,0鉄�$and $\mathbf{v}=鉄�2,4,0鉄�$.

$$\begin{gathered} \mathbf{u}脳\mathbf{v}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 1& 2& 0\\ 2& 4& 0\end{array}\right| \\ =0\mathbf{i}+0\mathbf{j}+0\mathbf{k} \end{gathered}$$

Now, calculate $(\mathbf{u}脳\mathbf{v})脳\mathbf{w}$

$$\begin{gathered} (\mathbf{u}脳\mathbf{v})脳\mathbf{w}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 0& 0& 0\\ -2& 6& 5\end{array}\right| \\ =0\mathbf{i}+0\mathbf{j}+0\mathbf{k} \end{gathered}$$

Hence, the value of $(\mathbf{u}脳\mathbf{v})脳\mathbf{w}$is$0\mathbf{i}+0\mathbf{j}+0\mathbf{k}$.

Consider vectors $\mathbf{v}=鉄�2,4,0鉄�$and $\mathbf{w}=鉄�4,8,0鉄�$

$$\begin{gathered} \mathbf{v}脳\mathbf{w}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 2& 4& 0\\ 4& 8& 0\end{array}\right| \\ =0\mathbf{i}+0\mathbf{j}+0\mathbf{k} \end{gathered}$$

Hence, the value of $\mathbf{v}脳\mathbf{w}$is $0\mathbf{i}+0\mathbf{j}+0\mathbf{k}$.

Now, calculate $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})$

$$\begin{gathered} \mathbf{u}脳(\mathbf{v}脳\mathbf{w})=\left|\begin{array}{lll}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 1& 2& 0\\ 0& 0& 0\end{array}\right| \\ =0\mathbf{i}+0\mathbf{j}+0\mathbf{k} \end{gathered}$$

Hence, the value of $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})$is $0\mathbf{i}+0\mathbf{j}+0\mathbf{k}$.

Therefore, for the vectors $\mathbf{u}=鉄�1,2,0鉄�,\mathbf{v}=鉄�2,4,0鉄�$and $\mathbf{w}=鉄�4,8,0鉄�$the result $(\mathbf{u}脳\mathbf{v})脳\mathbf{w}=\mathbf{u}脳(\mathbf{v}脳\mathbf{w})$ holds.