(a) Find the cross product of the two vectors overrightarrow (A)=8overrightarrow (i)+3overrightarrow (j) and overrightarrow (B)=-5overrightarrow (i)-7overrightarrow (j) (b) Verify explicitly that overrightarrow (A)times overrightarrow (B)=-overrightarrow (B)times overrightarrow (A)

(a) To find the cross product of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$, we use the determinant of a matrix formed by the unit vectors $\overrightarrow{i}$ and $\overrightarrow{j}$, and the components of the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$. The cross product is given by:<br /><br />$\overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix} \overrightarrow{i} & \overrightarrow{j} \\ 8 & 3 \\ -5 & -7 \end{vmatrix}$<br /><br />Expanding the determinant, we get:<br /><br />$\overrightarrow{A} \times \overrightarrow{B} = (3 \cdot -7 - (-7) \cdot 8)\overrightarrow{i} - (8 \cdot -7 - (-7) \cdot -5)\overrightarrow{j}$<br /><br />Simplifying the expression, we have:<br /><br />$\overrightarrow{A} \times \overrightarrow{B} = (-21 + 56)\overrightarrow{i} - (-56 + 35)\overrightarrow{j}$<br /><br />$\overrightarrow{A} \times \overrightarrow{B} = 35\overrightarrow{i} + 21\overrightarrow{j}$<br /><br />Therefore, the cross product of $\overrightarrow{A}$ and $\overrightarrow{B}$ is $35\overrightarrow{i} + 21\overrightarrow{j}$.<br /><br />(b) To verify that $\overrightarrow{A} \times \overrightarrow{B} = -\overrightarrow{B} \times \overrightarrow{A}$, we need to find the cross product of $\overrightarrow{B}$ and $\overrightarrow{A}$ and compare it with $-\overrightarrow{A} \times \overrightarrow{B}$.<br /><br />$\overrightarrow{B} \times \overrightarrow{A} = \begin{vmatrix} \overrightarrow{i} & \overrightarrow{j} \\ -5 & -7 \\ 8 & 3 \end{vmatrix}$<br /><br />Expanding the determinant, we get:<br /><br />$\overrightarrow{B} \times \overrightarrow{A} = (-7 \cdot 8 - 3 \cdot -5)\overrightarrow{i} - (-5 \cdot 3 - 3 \cdot 8)\overrightarrow{j}$<br /><br />Simplifying the expression, we have:<br /><br />$\overrightarrow{B} \times \overrightarrow{A} = (-56 + 15)\overrightarrow{i} - (-15 - 24)\overrightarrow{j}$<br /><br />$\overrightarrow{B} \times \overrightarrow{A} = -41\overrightarrow{i} + 39\overrightarrow{j}$<br /><br />Now, let's compare it with $-\overrightarrow{A} \times \overrightarrow{B}$:<br /><br />$-\overrightarrow{A} \times \overrightarrow{B} = -(35\overrightarrow{i} + 21\overrightarrow{j})$<br /><br />$-\overrightarrow{A} \times \overrightarrow{B} = -35\overrightarrow{i} - 21\overrightarrow{j}$<br /><br />We can see that $\overrightarrow{B} \times \overrightarrow{A} = -\overrightarrow{A} \times \overrightarrow{B}$, which verifies the given statement.