Finding the Volume of a Parallelepiped Using Vectors

The volume of a parallelepiped with adjacent sides represented by vectors A, B, and C is given by the magnitude of the scalar triple product of these vectors. The scalar triple product $A \cdot (B \times C)$ can be calculated as the determinant of the matrix formed by the components of the three vectors.

The given vectors are:

• A = 3i + j + k
• B = 5i + j – k
• C = i + j - k

We can write the components of these vectors as:

• A = $\begin{pmatrix} 3 \\ 1 \\ 1 \end{pmatrix}$
• B = $\begin{pmatrix} 5 \\ 1 \\ -1 \end{pmatrix}$
• C = $\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$

The volume (V) of the parallelepiped is given by the absolute value of the determinant of the matrix formed by these components:

$\text{V} = \left| \begin{vmatrix} 3 & 1 & 1 \\ 5 & 1 & -1 \\ 1 & 1 & -1 \end{vmatrix} \right|$

Now, let's calculate the determinant:

Determinant = $3 \begin{vmatrix} 1 & -1 \\ 1 & -1 \end{vmatrix} - 1 \begin{vmatrix} 5 & -1 \\ 1 & -1 \end{vmatrix} + 1 \begin{vmatrix} 5 & 1 \\ 1 & 1 \end{vmatrix}$

Determinant = $3((1)(-1) - (-1)(1)) - 1((5)(-1) - (-1)(1)) + 1((5)(1) - (1)(1))$

Determinant = $3(-1 + 1) - 1(-5 + 1) + 1(5 - 1)$

Determinant = $3(0) - 1(-4) + 1(4)$

Determinant = $0 + 4 + 4$

Determinant = $8$

The volume of the parallelepiped is the absolute value of the determinant:

Volume = $|8| = 8$

Thus, the volume of the parallelepiped with the given adjacent side vectors is 8 cubic units.

Revision Table: Key Vector Concepts

Additional Information: Geometric Interpretation of Scalar Triple Product

The absolute value of the scalar triple product $|A \cdot (B \times C)|$ has a clear geometric meaning: it is the volume of the parallelepiped defined by the three vectors A, B, and C when they share the same initial point. The cross product $B \times C$ gives a vector whose magnitude is the area of the parallelogram formed by B and C, and whose direction is perpendicular to the plane containing B and C. The dot product of A with this cross product ($A \cdot (B \times C)$) is then the product of the magnitude of A, the magnitude of $B \times C$ (area of the base), and the cosine of the angle between A and $B \times C$. Since $B \times C$ is perpendicular to the base plane, the cosine of the angle relates A's height above the base to its magnitude. Essentially, $A \cdot (B \times C)$ is (Area of base parallelogram formed by B and C) $\times$ (Height of the parallelepiped relative to that base, which is the component of A orthogonal to the base). The sign of the scalar triple product indicates the orientation of the vectors (whether they form a right-handed or left-handed system).