Application of inner product and cross product

Exercise1.5

1.

Show that $\boldsymbol{A} = :\boldsymbol{i} + 2\:\boldsymbol{j} - \:\boldsymbol{k}, \bolds... ...{k}, \boldsymbol{C} = -\:\boldsymbol{i} + 3\:\boldsymbol{j} + 4\:\boldsymbol{k}$ is not coplanar.

2.

For ${\bf A} = {\bf i} + 2{\bf j} + {\bf k}, {\bf B} = 2{\bf i} - {\bf j} + {\bf k}, {\bf C} = -{\vert bf i} + {\bf j} + 2{\bf k}$, find ${\bf A} \cdot ({\bf B} \times {\bf C}$

3.

Let ${\bf A} = A_1 {\bf i} + A_2{\bf j} + A_3{\bf k}, {\bf B} = B_1 {\bf i} + B_2{\bf j} + B_3{\bf k}$. Then show that $\vert\boldsymbol{A}\times \boldsymbol{B}\vert^2 = (\boldsymbol{A}\cdot\boldsymbol{A})(\boldsymbol{B} \cdot\boldsymbol{B}) - (\boldsymbol{A}\cdot\boldsymbol{B})^2$．