Vector Parallelogram Area Calculation

This explanation covers how to find the area of a parallelogram when its sides are given as vectors. We utilize the vector cross product method for this calculation.

Defining the Vectors

Let the two vectors representing the adjacent sides of the parallelogram be \(\vec{a}\) and \(\vec{b}\).

• Vector \(\vec{a}\) = \(\hat{i} + 2\hat{j} + 3\hat{k}\)
• Vector \(\vec{b}\) = \(2\hat{i} + \hat{j} + 2\hat{k}\)

We can express these vectors in component form:

• \(\vec{a} = \langle 1, 2, 3 \rangle\)
• \(\vec{b} = \langle 2, 1, 2 \rangle\)

Parallelogram Area Formula with Vectors

The area (\(A\)) of a parallelogram formed by two vectors \(\vec{a}\) and \(\vec{b}\) originating from the same point is equal to the magnitude of their cross product (\(\vec{a} \times \vec{b}\)):

A = $|\vec{a} \times \vec{b}|

Cross Product Calculation

First, we compute the cross product \(\vec{a} \times \vec{b}\) using the determinant formula:

\(\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 2 & 1 & 2 \end{vmatrix}\)

Expanding the determinant along the first row:

• \(\hat{i}\) component: \((2 \times 2 - 3 \times 1) = (4 - 3) = 1\)
• \(\hat{j}\) component: \(-(1 \times 2 - 3 \times 2) = -(2 - 6) = -(-4) = 4\)
• \(\hat{k}\) component: \((1 \times 1 - 2 \times 2) = (1 - 4) = -3\)

Therefore, the cross product vector is:

\(\vec{a} \times \vec{b} = 1\hat{i} + 4\hat{j} - 3\hat{k}\)

Or, in component notation: \(\vec{a} \times \vec{b} = \langle 1, 4, -3 \rangle\)

Magnitude Calculation for Area

Next, we find the magnitude of the resulting cross product vector \(\langle 1, 4, -3 \rangle\). The magnitude of a vector \(\vec{v} = \langle x, y, z \rangle\) is given by the formula \(|\vec{v}| = \sqrt{x^2 + y^2 + z^2}\).

Applying this formula to our cross product vector:

A = \(|\langle 1, 4, -3 \rangle| = \sqrt{1^2 + 4^2 + (-3)^2}\)

A = \(\sqrt{1 + 16 + 9}\)

A = \(\sqrt{26}\)

Final Area Result

The area of the parallelogram is \(\sqrt{26}\) square units.