Determining the Angle Between Two Sides of a Triangle Using Vectors in 3D

Imagine a class of schoolchildren are asked by their teacher to “Construct a triangle which has two sides of length let’s say 6cm and 7cm”. The result there are likely to be as many different triangles as there are class-members! We need more information to be specific.

You cannot uniquely define a triangle with just two sides. You need three independent pieces of information to define a triangle, such as three side lengths or two side lengths and the included angle. Similarly, knowing only the lengths of two sides does not provide enough information to determine a specific angle.

Why Do Two Sides Not Determine the Angle?

With just two sides of a triangle, an infinite number of triangles can be formed, varying by the included angle. Therefore, knowing only the lengths of two sides is insufficient to determine the angle they form with each other. If you need the angle, additional information such as the included angle or the third side is required.

Calculation Using Vectors

Since two sides alone cannot determine the angle, how can we find the angle between the two sides of a triangle in 3D space? Let's use vectors. To calculate the angle between two sides of a triangle using vectors, first, you need to identify the vectors corresponding to the sides of the triangle. Then, you can apply the dot product formula to find the angle.

Steps to Find the Angle Using Vectors

Find the vectors of the sides: Suppose the vertices of the triangle are $A$, $B$, and $C$. The vectors representing the sides of the triangle could be $overrightarrow{AB}$ and $overrightarrow{AC}$. This is the first step in identifying the vectors.

Use the dot product formula: The dot product of two vectors $overrightarrow{u}$ and $overrightarrow{v}$ is given by $overrightarrow{u} cdot overrightarrow{v} |overrightarrow{u}||overrightarrow{v}|cos(theta)$, where $theta$ is the angle between the vectors. Therefore, if $overrightarrow{u} overrightarrow{AB}$ and $overrightarrow{v} overrightarrow{AC}$, then $overrightarrow{AB} cdot overrightarrow{AC} |overrightarrow{AB}||overrightarrow{AC}|cos(theta)$.

Solve for the angle $theta$: Rearrange the dot product formula to solve for $theta$. $cos(theta) frac{overrightarrow{AB} cdot overrightarrow{AC}}{|overrightarrow{AB}||overrightarrow{AC}|}$. Then, find the inverse cosine (arccos) to get the angle $theta$.

For example, if $overrightarrow{AB} (a_1, a_2, a_3)$ and $overrightarrow{AC} (b_1, b_2, b_3)$, their dot product is $overrightarrow{AB} cdot overrightarrow{AC} a_1b_1 a_2b_2 a_3b_3$. The magnitudes of the vectors are $|overrightarrow{AB}| sqrt{a_1^2 a_2^2 a_3^2}$ and $|overrightarrow{AC}| sqrt{b_1^2 b_2^2 b_3^2}$. Plugging these into the formula, we can calculate the angle $theta$ between the two vectors, which represents the angle between the two sides of the triangle.

Conclusion

In conclusion, two sides of a triangle are insufficient to determine a unique angle. However, by using vector methods, we can calculate the angle between two sides of a triangle in 3D space. This method requires finding the vectors corresponding to the sides of the triangle and then applying the dot product formula. With this detailed approach, educators and students can better understand the geometric relationships within triangles.