When i started to learn vector mathematics , i had found many methods to determine whether a point is inside a triangle or not.Easy method is not the fastest. This type of calculations are very important for making a fast graphics library.
Following are the methods which i remember now.
1. Using Cross Products
Take cross product between each triangle edge and the point to check. If the direction of cross product result is for the three edges that point is inside the triangle , otherwise not
2. Using the angle between the vectors made by point and triangle coordinates.If the sum of angle is 360 the point is inside , otherwise not.
3. Using Point and Plane test.See the figure. You need to know plane equation to understand it.
4. Using Bary centric Coordinates
This is the fastest of above. It involves checking the point in Bary centric coordinates system.