## Linear Independence and Volumes of Parallelepipeds

If a parallelogram is defined by two vectors and then the area of the parallelogram is defined by ( Note where is the angle between and  If the vectors and are joined by a third vector to 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 by and ) multiplied by the height. is perpendicular to both and so is in the direction of the vertical height. By taking the dot product of with and dividing by we obtain the component of perpendicular to This is the height of the parallelpiped. Multiplying by gives the volume: This is illustrated below.  If the vectors are 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. 