Question

Find a vector orthogonal to the given vectors. $$\langle 0,1,2\rangle \text { and }\langle-2,0,3\rangle$$

Short answer

Answer: The orthogonal vector is $\langle 3, -4, 2 \rangle$.

Step-by-step solution

Recall the cross product formula

To compute the cross product of two vectors, say \(\vec{A} = \langle A_x, A_y, A_z\rangle\) and \(\vec{B} = \langle B_x, B_y, B_z\rangle\), we have the following formula for their cross product \(\vec{C}\): $$\vec{C} = \langle A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x \rangle.$$

Substitute the given vectors into the cross product formula

Substitute the given vectors \(\vec{A} = \langle 0,1,2\rangle\) and \(\vec{B} = \langle -2,0,3\rangle\) into the cross product formula: \begin{align*} \vec{C} &= \langle (1)(3) - (2)(0), (2)(-2) - (0)(3), (0)(0) - (1)(-2) \rangle \\ &= \langle 3, -4, 2 \rangle. \end{align*}

Conclusion

The vector orthogonal to the given vectors \(\langle 0,1,2\rangle\) and \(\langle -2,0,3\rangle\) is \(\langle 3, -4, 2 \rangle\).