Question

Show that a x b is perpendicular to both a and b where a= -5i+6j-3k and b= 7i+8j+4k

Answer 1

The cross-product is:

$$\mathbf{c}=\mathbf{a}\times \mathbf{b} = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ -5 & 6 & -3 \\ 7 & 8 & 4 \end{vmatrix}$$

where $\mathbf{i},\mathbf{j},\mathbf{k}$ are unit vecrots of Cartesian coordinates. Thus, obtain:

$$\mathbf{c} = 48\mathbf{i}-\mathbf{j} -82 \mathbf{k}$$

To check the perpendicularity, let's find dot products:

$$\mathbf{a}\cdot \mathbf{c} = (-5)\cdot 48 +6\cdot (-1) + (-3)\cdot (-82) = 0\\ \mathbf{b}\cdot \mathbf{c} = 7\cdot 48 +8\cdot (-1) + 4\cdot (-82) = 0$$

Thus, a x b is perpendicular to both a and b.