Linear Independence and Volumes of Parallelepipeds
If a parallelogram is defined by two vectorsandthen the area of the parallelogram is defined by( Notewhereis the angle betweenand
If the vectorsandare joined by a third vectorto form a solid shape, then the volume of the solid is the area of the base (which we may consider to be the area of the parallelepiped formed byand) multiplied by the height.is perpendicular to bothandso is in the direction of the vertical height. By taking the dot product ofwithand dividing bywe obtain the component ofperpendicular toThis is the height of the parallelepiped. Multiplying bygives the volume: This is illustrated below.
If the vectorsare in the same plane, then they are linearly dependent, since three vectors in a two dimensional space are linearly dependent. They all lie in the same plane and the height of the parallelepiped is zero. If a matrix is formed with the columns or rows consisting of the three vectors, the determinant of this matrix will be zero since the vectors are linearly dependent.