Cross product definition

Study guide and 12 practice problems on:

The cross product of $\mathbf{x} = \langle x_1, x_2, x_3\rangle$ and $\mathbf{y} = \langle y_1, y_2, y_3\rangle$ is

\begin{align}\mathbf{x} \times \mathbf{y} &= \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{vmatrix}\\ \ \\ &= \begin{vmatrix}x_2 & x_3 \\ y_2 & y_3 \end{vmatrix} \ \mathbf{i} - \begin{vmatrix}x_1 & x_3 \\ y_1 & y_3 \end{vmatrix} \ \mathbf{j} + \begin{vmatrix}x_1 & x_2 \\ y_1 & y_2 \end{vmatrix} \ \mathbf{k} \end{align}

Practice problems

Find a vector perpendicular to $\langle1, 2, 3 \rangle$.
Solution
Using a cross product, find a vector in 2d that is 90 degrees counterclockwise from $\langle a, b\rangle$.
Solution
Show that $\bfa \times \bfb$ is perpendicular to $\bfa$ by computing a dot product.
Solution
Compute $\bfi \times (\bfi + \bfk)$ in two ways:
By the determinant formula
By expanding the sum and recalling the cross products of standard coordinate vectors with each other
Solution
What is the relationship of $\bfa \times \bfb$ and $\bfb \times \bfa$? Prove it using components.
Solution
Let $\bfa = \langle a_1, a_2, a_3 \rangle, \bfb = \langle b_1, b_2, b_3\rangle, \bfc = \langle c_1, c_2, c_3 \rangle.$ Show that the triple product of the vectors equals the determinant of a matrix whose rows is given by the vectors:$$\bfa \cdot (\bfb \times \bfc) = \begin{vmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$$
Solution
Find the area of the triangle in 3-space between $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$.
Solution
Find a unit vector perpendicular to $\langle 1, 1, 1\rangle$ and $\langle 1, 0, 1 \rangle.$
Solution
Consider the triangle in three-space given by $\langle a, 0, 0\rangle, \langle 0, b, 0 \rangle, \langle 0, 0, c \rangle.$ Find a vector that is perpendicular to the triangle and has length equal to the area of the triangle.
Solution
Sketch the parallelogram spanned by $\langle 1,1\rangle$ and $\langle 0, 1\rangle$. Use a cross product to find its area.
Solution
more problems

Cross product definition

Related topics

Practice problems