1. If a=2i-3j-k and b=i+4j-2k Show that overrightarrow (a)times overrightarrow (b) is not equal to bxa

To show that $\overrightarrow{a} \times \overrightarrow{b}$ is not equal to $\overrightarrow{b} \times \overrightarrow{a}$, we need to calculate both cross products and compare them.<br /><br />Given:<br />$\overrightarrow{a} = 2i - 3j - k$<br />$\overrightarrow{b} = i + 4j - 2k$<br /><br />First, let's calculate $\overrightarrow{a} \times \overrightarrow{b}$:<br /><br />$\overrightarrow{a} \times \overrightarrow{b} = \begin{vmatrix} i & j & k \\ 2 & -3 & -1 \\ 1 & 4 & -2 \end{vmatrix} = i((-3)(-2) - (-1)(4)) - j((2)(-2) - (-1)(1)) + k((2)(4) - (-3)(1))$<br /><br />Simplifying the above expression, we get:<br /><br />$\overrightarrow{a} \times \overrightarrow{b} = i(6 + 4) - j(-4 + 1) + k(8 + 3) = 10i + 3j + 11k$<br /><br />Now, let's calculate $\overrightarrow{b} \times \overrightarrow{a}$:<br /><br />$\overrightarrow{b} \times \overrightarrow{a} = \begin{vmatrix} i & j & k \\ 1 & 4 & -2 \\ 2 & -3 & -1 \end{vmatrix} = i((4)(-1) - (-2)(-3)) - j((1)(-1) - (-2)(2)) + k((1)(-3) - (4)(2))$<br /><br />Simplifying the above expression, we get:<br /><br />$\overrightarrow{b} \times \overrightarrow{a} = i(-4 - 6) - j(-1 + 4) + k(-3 - 8) = -10i + 3j - 11k$<br /><br />Comparing the two results, we can see that $\overrightarrow{a} \times \overrightarrow{b}$ is not equal to $\overrightarrow{b} \times \overrightarrow{a}$, as they have opposite signs in the $i$ and $k$ components.