This page will be about finding areas of parallelograms and triangles using vectors, norms and the cross product.
Table Of Contents
- Geometric Intuition
- Deriving The Area Of A Parallelogram Using Cross Products
- The Area Of The Parallelogram
- The Area Of A Triangle
- Practice Problems
Suppose we have two vectors in three space as shown in the picture below (I use vectors and while the picture uses and .
In the vector setting, distances are taken by using norms. In this case, we would have the norms and for vectors and respectively.
On one side of the parallelogram (the left side in this case), we draw an altitude line to the base which forms a triangle. In this triangle, the norm is the hypotenuse and the angle (with degrees) is the angle of the parallelogram and the triangle.
We seek to find the side length opposite to the angle . Using trigonometry (SOH CAH TOA) we have:
Deriving The Area Of A Parallelogram Using Cross Products
Recall that one of the properties of cross products and dot products is Lagrange’s identity.
In addition, given an angle with degrees we have the dot product equation
In Lagrange’s identity, we substitute with the above equation, factor and the equation (or ) to get:
From , we can take the square root of both sides to obtain
The Area Of The Parallelogram
The area of a parallelogram is the product of the base length () times the altitude. This formula can be expressed as:
The above formula can also be expressed as (because of the derivation earlier).
The Area Of A Triangle
It is known that the area of a triangle is half the area of a paraellogram. Using cross products and norms, the formula for the area of a triangle is:
Given the vectors and , Find the area of the parallelogram enclosed by these two vectors.
Given the two vectors and , we find the cross product first. The norm of this cross product will be calculated to obtain the area of the parallelogram enclosed by the two vectors.
One can show that the cross product is .
Taking the norm of this product yields:
Consider the points , and . Find the area of the triangle determined by the three points.
The strategy is to create two vectors from the three points, find the cross product of the two vectors and then take the half the norm of the cross product.
One vector is . The second vector would be
One can show that the cross product of the vectors and is .
The norm of this cross product computes to:
The area of a triangle is half the norm of the cross product. It computes to:
Here are few practice problems. Solutions are in the next section.
1) Given the vectors and , Find the area of the parallelogram enclosed by these two vectors.
2) Given the vectors and , Find the area of the triangle enclosed by these two vectors.
3) Find the area of the triangle determined by the points , and .
1) . The area of the parallelogram computes to square units.
2) . The area of the triangle computes to square units.
3) , ; ; square units.
Elementary Linear Algebra (Tenth Edition) by Howard Anton