Cross produc

Definition 1..1  

In the two space vectors $\boldsymbol{A}, \boldsymbol{B}$, the size is the area of ?the parallelogram created by $\boldsymbol{A}, \boldsymbol{B}$ $\vert\boldsymbol {A }\vert \boldsymbol{B}\vert \sin{\ theta}$, the direction is perpendicular to both $\boldsymbol{A}, \boldsymbol{B}$, and $\boldsymbol{A}$ is rotating within $180^{\ circ}$ and overlapping in the direction of $\boldsymbol{B}$, the vector determined as the direction of the right-handed screw is the cross product of $\boldsymbol{A}, \boldsymbol{B}$ ( cross product) is called $\boldsymbol{A} \times \boldsymbol{B}$．For $\boldsymbol{A} = {\bf0}, \boldsymbol{B} = {\bf0}, \theta = 0$.

$$(A_{1}\:\boldsymbol{i} + A_{2}\:\boldsymbol{j} + A_{3}\:\boldsymb... ...{1} - A_{1}B_{3})\:\boldsymbol{j} + (A_{1}B_{2} - A_{2}B_{1})\:\boldsymbol{k}.$$

Using the determinant, we can express the right-hand side as follows:

$$(A_{1}\:\boldsymbol{i} + A_{2}\:\boldsymbol{j} + A_{3}\:\boldsymb... ...ldsymbol{k}\\ A_{1}&A_{2}&A_{3}\\ B_{1}&B_{2}&B_{3} \end{array}\right\vert .$$

Example 1..3  

For $\boldsymbol{A} = {}^t[1\ 2\ 1], \boldsymbol{B} = {}^t[2\ -1\ -2]$，find $\boldsymbol{A} \times \boldsymbol{B}$．

Answer $\left\vert\begin{array}{rrr} \:\boldsymbol{i} & \:\boldsymbol{j} & \:\boldsymbo... ...= -3\:\boldsymbol{i} + 4\:\boldsymbol{j} + -5\:\boldsymbol{k} = {}^t[-3\ 4\ -5]$

Question 1..4  

For $\boldsymbol{A} = 2\:\boldsymbol{i} - 3\:\boldsymbol{j} - \:\boldsymbol{k},\ \boldsymbol{B} = \:\boldsymbol{i} + 4\:\boldsymbol{j} - 2\:\boldsymbol{k}$，find the following vectors:． (1) $\boldsymbol{A} \times \boldsymbol{B}$

(2) $(2\boldsymbol{A} - 3\boldsymbol{B}) \times (\boldsymbol{A} + 2\boldsymbol{B})$

Question 1..5  

Let $\boldsymbol{A} = \:\boldsymbol{i} + 2\:\boldsymbol{j} + \:\boldsymbol{k}, \boldsymbol{B} = 2\:\boldsymbol{i} - \:\boldsymbol{j} - 2\:\boldsymbol{k}$．

(1) Find the area of the parallelogram when these are regarded as two adjacent sides． (2) Find a vector other than the zero vector that is orthogonal to these．

Question 1..6  

Prove the followings:．

$$\vert\boldsymbol{A} \times \boldsymbol{B}\vert^2 = (\boldsymbol{A... ...l{A})(\boldsymbol{B}\cdot\boldsymbol{B})-(\boldsymbol{A} \cdot\boldsymbol{B})^2$$