Problem on cross products and the area of a parallelogram - Leading Lesson

Problem on cross products and the area of a parallelogram

Sketch the parallelogram spanned by $\langle 1,1\rangle$ and $\langle 0, 1\rangle$. Use a cross product to find its area.

Solution
We begin by sketching the vectors:
The parallelogram spanned by these vectors is:
Recall that
Cross product and area of parallelograms

The area of the parallelogram spanned by $\bfx$ and $\bfy$ is $\left | \bfx \times \bfy \right |$.
Thus, we need to take the cross product of $\langle 1, 1\rangle$ with $\langle 1, 0 \rangle$.
Because the cross product requires vectors to be three-dimensional, we must view these two-dimensional vectors as having a $\bfk$ component of zero: $\langle 1, 1, 0\rangle$ and $\langle 1, 0, 0\rangle$.
Recall that
Cross product definition

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} \end{align}
Determinant of a 3x3 matrix

Determinant of a 2x2 matrix

Using $\bfx = \langle 1, 1, 0\rangle$ and $\bfy = \langle 1, 0, 0\rangle$, we get \begin{align}\mathbf{x} \times \mathbf{y} &= \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{vmatrix}\\ &= \begin{vmatrix}1 & 0 \\ 0 & 0 \end{vmatrix} \ \mathbf{i} - \begin{vmatrix}1 & 0 \\ 1 & 0 \end{vmatrix} \ \mathbf{j} + \begin{vmatrix}1 & 1 \\ 1 & 0 \end{vmatrix} \ \mathbf{k} \\ &= 0 \ \bfi + 0 \ \bfj - \bfk \end{align}
Definition of vector length

The length of $\bfx \times \bfy$ is $\left | -\bfk \right| = 1$.
The area of the parallelogram is $1$.

Related topics

Multivariable calculus
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