91影视

(a) Take the vectors and .

The goal is to demonstrate that $\mathbf{i}脳(\mathbf{v}脳\mathbf{w})=w_1\mathbf{v}-v_1\mathbf{w}$.

Find the cross product$\mathrm{i}脳(\mathrm{v}脳\mathbf{w})$ to prove the result.

The by-product is$(\mathbf{v}脳\mathbf{W})$ stands for:

$\mathbf{v}脳\mathbf{w}=$$\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ v_1& v_2& v_3\\ w_1& w_2& w_3\end{array}\right|$

$=\left(v_2{w}_3-v_3{w}_2\right)\mathbf{i}+\left(v_3{w}_1-v_1{w}_3\right)\mathbf{j}+\left(v_1{w}_2-v_2{w}_1\right)\mathbf{k}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right)$

The by-product $\mathbf{i}脳(\mathbf{v}脳\mathbf{w})$is

$\mathbf{i}脳(\mathbf{v}脳\mathbf{w})=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 1& 0& 0\\ v_2{w}_3-v_3{w}_2& v_3{w}_1-v_1{w}_3& v_1{w}_2-v_2{w}_1\end{array}\right|\text{(Using 1)}$

$=0\mathbf{i}-\left(v_1{w}_2-v_2{w}_1\right)\mathbf{j}+\left(v_3{w}_1-v_1{w}_3\right)\mathbf{k}$

The expression$w_1\mathbf{v}-v_1\mathbf{w}$ yields:

The following are the results of the equations (1) and (2) $\mathbf{i}脳(\mathbf{v}脳\mathbf{w})=w_1\mathbf{v}-v_1\mathbf{w}$

(b) Consider the vectors and .

The goal is to calculate $\mathbf{j}脳(\mathbf{v}脳\mathbf{w})\text{and}\mathbf{k}脳(\mathbf{v}脳\mathbf{w})$.

Use the result $\mathbf{i}脳(\mathbf{v}脳\mathbf{w})=w_1\mathbf{v}-v_1\mathbf{w}$to determine the value of $\mathbf{j}脳(\mathbf{v}脳\mathbf{w})$to get the value of $j脳(v脳w)$.

The value of $\mathbf{j}脳(\mathbf{v}脳\mathbf{w})$in generalizing the result $\mathbf{i}脳(\mathbf{v}脳\mathbf{W})={\mathbf{w}}_1\mathbf{v}-v_1\mathbf{W}$is:

$j脳(\mathbf{v}脳\mathbf{w})=w_2\mathbf{v}-v_2\mathbf{w}$

As a result ,$\mathbf{j}脳(\mathbf{v}脳\mathbf{w})\text{is}{w}_2\mathbf{v}-v_2\mathbf{w}$

The value of $\mathbf{k}脳(\mathbf{v}脳\mathbf{w})$when generalising the result $\mathbf{i}脳(\mathbf{v}脳\mathbf{w})=w_1\mathbf{v}-v_1\mathbf{w}$is:

$\mathbf{k}脳(\mathbf{v}脳\mathbf{w})=w_3\mathbf{v}-v_3\mathbf{w}$

As a result,$\mathbf{k}脳(\mathbf{v}脳\mathbf{w})\text{is}{\mathbf{w}}_3\mathbf{v}-v_3\mathbf{w}$

(c) The goal is to demonstrate that $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})=(\mathbf{u}路\mathbf{w})\mathbf{v}-(\mathbf{u}路\mathbf{v})\mathbf{w}$.

Use the following relationships to show your point.

$\mathbf{i}脳(\mathbf{v}脳\mathbf{w})=w_1\mathbf{v}-v_1\mathbf{w}$

$\mathbf{j}脳(\mathbf{v}脳\mathbf{w})=w_2\mathbf{v}-v_2\mathbf{w}$

$\mathbf{k}脳(\mathbf{v}脳\mathbf{w})={\mathbf{w}}_3\mathbf{v}-v_3\mathbf{w}$

The expression$\mathbf{i}脳(\mathbf{v}脳\mathbf{w})=w_1\mathbf{v}-v_1\mathbf{w}$ can be expressed as follows: $\mathbf{i}脳(\mathbf{v}脳\mathbf{w})=(\mathbf{i}路\mathbf{w})\mathbf{v}-(\mathbf{i}路\mathbf{v})\mathbf{w}$(Because$\mathbf{i}路\mathbf{i}=1,\mathbf{i}路\mathbf{j}=0,\mathbf{i}路\mathbf{k}=0)$ As a result of generalising the result, the following equation emerges.$\mathbf{u}脳(\mathbf{v}脳\mathbf{w})=(\mathbf{u}路\mathbf{w})\mathbf{v}-(\mathbf{u}路\mathbf{v})\mathbf{w}$

The result is proven.

(d) The goal is to calculate the result of $(\mathbf{u}脳\mathbf{v})脳\mathbf{w}$

Use the result $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})=(\mathbf{u}路\mathbf{w})\mathbf{v}-(\mathbf{u}路\mathbf{v})\mathbf{w}$and the anti-commutativity of the vectors to get the result.

Due to the fact that the vectors are not commutative.

As a result,$(\mathbf{u}脳\mathbf{v})脳\mathbf{w}=-\mathbf{u}脳(\mathbf{v}脳\mathbf{w})$

As a result,$(\mathbf{u}脳\mathbf{v})脳\mathbf{w}=-\left[\right(\mathbf{u}路\mathbf{w})\mathbf{v}-(\mathbf{u}路\mathbf{v}\left)\mathbf{w}\right]$(Substitution) $=(\mathbf{u}路\mathbf{v})\mathbf{w}-(\mathbf{u}路\mathbf{w})\mathbf{v}$

As a result,$(\mathbf{u}脳\mathbf{v})脳\mathbf{w}\text{is}(\mathbf{u}路\mathbf{v})\mathbf{w}-(\mathbf{u}路\mathbf{w})\mathbf{v}$

(e)

The goal is to show that cross-product is not synonymous with associative.

The value of $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})$is : $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})=(\mathbf{u}路\mathbf{w})\mathbf{v}-(\mathbf{u}路\mathbf{v})\mathbf{w}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right)$

$(\mathbf{u}脳\mathbf{v})脳\mathbf{w}=(\mathbf{u}路\mathbf{v})\mathbf{w}-(\mathbf{u}路\mathbf{w})\mathbf{v}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(2\right)$has the following value:

From equation $\left(1\right)$and $\left(2\right)$

$\mathbf{u}脳(\mathbf{v}脳\mathbf{w})鈮�(\mathbf{u}脳\mathbf{v})脳\mathbf{w}$

As a result, cross-product is not an associative term.

(f)

The goal is to determine whether $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})=(\mathbf{u}脳\mathbf{v})脳\mathbf{w}$

Only when $(\mathbf{u}路\mathbf{w})\mathbf{v}-(\mathbf{u}路\mathbf{v})\mathbf{w}=(\mathbf{u}路\mathbf{v})\mathbf{w}-(\mathbf{u}路\mathbf{w})\mathbf{v}$

$2\left[\right(\mathbf{u}路\mathbf{w})\mathbf{v}-(\mathbf{u}路\mathbf{v}\left)\mathbf{w}\right]=0$

$(\mathbf{u}路\mathbf{w})\mathbf{v}-(\mathbf{u}路\mathbf{v})\mathbf{w}=0$
$(\mathbf{u}-\mathbf{w})\mathbf{v}=\mathbf{0}$and $\mathbf{u}路\mathbf{v}=0$

$\mathbf{u}路\mathbf{w}=0$and $\mathbf{u}路\mathbf{v}=0$