The diagonals of a parallelogram can be represented by vectors, and the area of the parallelogram can be efficiently calculated using the cross product of these diagonal vectors.
In a parallelogram, if the adjacent sides are represented by vectors $\mathbf{a}$ and $\mathbf{b}$, the diagonals can be represented by the vectors:
- $\mathbf{d}_1 = \mathbf{a} + \mathbf{b}$ (representing one diagonal)
- $\mathbf{d}_2 = \mathbf{b} - \mathbf{a}$ (representing the other diagonal)
These diagonal vectors connect opposite vertices of the parallelogram.
Calculating Parallelogram Area Using Diagonal Vectors
While the area of a parallelogram can be found using the magnitude of the cross product of two adjacent side vectors (Area = $|\mathbf{a} \times \mathbf{b}|$), an alternative and useful formula utilizes the diagonal vectors.
As referenced, the area of a parallelogram can be found using its diagonal vectors, $\mathbf{d}_1$ and $\mathbf{d}_2$, with the formula:
Area = $\frac{1}{2} |(\mathbf{d}_1 \times \mathbf{d}_2)|$
This formula states that the area is half the magnitude of the cross product of the two diagonal vectors.
Why Half the Magnitude?
The cross product $\mathbf{d}_1 \times \mathbf{d}_2$ results in a vector whose magnitude is equal to the area of the parallelogram formed by $\mathbf{d}_1$ and $\mathbf{d}_2$ as adjacent sides. This parallelogram (often called the diagonal parallelogram) is different from the original parallelogram. The area of the original parallelogram is precisely half the area of this larger parallelogram formed by the diagonals.
Example Calculation
Let's illustrate how to find the area of a parallelogram given its diagonal vectors.
Suppose the diagonal vectors are given as:
- $\mathbf{d}_1 = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k}$
- $\mathbf{d}_2 = \mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$
Steps to Find the Area:
-
Calculate the Cross Product: Find the cross product $\mathbf{d}_1 \times \mathbf{d}_2$.
$\mathbf{d}_1 \times \mathbf{d}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 3 & 1 & -2 \ 1 & -3 & 4 \end{vmatrix}$
$= \mathbf{i}((1)(4) - (-2)(-3)) - \mathbf{j}((3)(4) - (-2)(1)) + \mathbf{k}((3)(-3) - (1)(1))$
$= \mathbf{i}(4 - 6) - \mathbf{j}(12 - (-2)) + \mathbf{k}(-9 - 1)$
$= -2\mathbf{i} - 14\mathbf{j} - 10\mathbf{k}$ -
Find the Magnitude of the Cross Product: Calculate the magnitude of the resulting vector.
$|\mathbf{d}_1 \times \mathbf{d}_2| = \sqrt{(-2)^2 + (-14)^2 + (-10)^2}$
$= \sqrt{4 + 196 + 100}$
$= \sqrt{300}$
$= \sqrt{100 \times 3}$
$= 10\sqrt{3}$ -
Apply the Area Formula: Use the formula Area = $\frac{1}{2} |(\mathbf{d}_1 \times \mathbf{d}_2)|$.
Area $= \frac{1}{2} \times 10\sqrt{3}$
Area $= 5\sqrt{3}$
So, the area of the parallelogram with these diagonal vectors is $5\sqrt{3}$ square units.
Here is a summary of the steps in a table:
| Step | Calculation | Result |
|---|---|---|
| Diagonal Vectors | $\mathbf{d}_1 = \langle 3, 1, -2 \rangle$, $\mathbf{d}_2 = \langle 1, -3, 4 \rangle$ | - |
| Calculate Cross Product $\mathbf{d}_1 \times \mathbf{d}_2$ | $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 3 & 1 & -2 \ 1 & -3 & 4 \end{vmatrix} = -2\mathbf{i} - 14\mathbf{j} - 10\mathbf{k}$ | $\langle -2, -14, -10 \rangle$ |
| Magnitude of Cross Product | $\sqrt{(-2)^2 + (-14)^2 + (-10)^2} = \sqrt{300}$ | $10\sqrt{3}$ |
| Apply Area Formula (1/2 * Magnitude) | $\frac{1}{2} \times 10\sqrt{3}$ | $5\sqrt{3}$ |
Key Takeaways
- The diagonals of a parallelogram can be represented as vectors derived from its adjacent sides ($\mathbf{d}_1 = \mathbf{a} + \mathbf{b}$, $\mathbf{d}_2 = \mathbf{b} - \mathbf{a}$).
- The area of a parallelogram can be calculated if the diagonal vectors $\mathbf{d}_1$ and $\mathbf{d}_2$ are known.
- The formula for the area using diagonal vectors is Area = $\frac{1}{2} |(\mathbf{d}_1 \times \mathbf{d}_2)|$, as stated in the reference.
Using diagonal vectors provides an alternative approach to finding the area, complementing the method using adjacent side vectors.
