Vector Triple Product | Scalar Triple Product | Proof of Vector Triple Product | Vector Triple Product Example | Scalar Triple Product Proof | Scalar Triple Product Example

 

Vector Triple Product | Scalar Triple Product

Vector Triple Product

Definition:

The vector triple product involves three vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. It is defined as:

$$\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$$

Expansion:

To simplify $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$, use the vector triple product identity:

$$\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$$

Proof:

1. Express Cross Product in Component Form:

   Using the definition of the cross product in component form:

   $$\mathbf{b}\times \mathbf{c}=\mid \begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\mid$$

   Suppose:

   $$\mathbf{b} \times \mathbf{c} = (b_2c_3 - b_3c_2, b_3c_1 - b_1c_3, b_1c_2 - b_2c_1)$$

2. Cross Product with $\mathbf{a}$:

   Now compute $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$:

   $$\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\mid \begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_1& a_2& a_3\\ b_2{c}_3-b_3{c}_2& b_3{c}_1-b_1{c}_3& b_1{c}_2-b_2{c}_1\end{array}\mid$$

   Expanding this determinant, we get:

   $$\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (a_2 (b_1c_2 - b_2c_1) - a_3 (b_1c_1 - b_3c_2), a_3 (b_1c_3 - b_2c_1) - a_1 (b_1c_3 - b_3c_1), a_1 (b_2c_2 - b_2c_1) - a_2 (b_1c_2 - b_2c_1))$$

3. Simplify Using Dot Products:

   To simplify, use the dot products:

   $$\mathbf{a} \cdot \mathbf{c} = a_1c_1 + a_2c_2 + a_3c_3$$ $$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$$

   Thus:

   $$(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} = (a_1c_1 + a_2c_2 + a_3c_3) (b_1, b_2, b_3)$$ $$(\mathbf{a} \cdot \mathbf{b}) \mathbf{c} = (a_1b_1 + a_2b_2 + a_3b_3) (c_1, c_2, c_3)$$

   The resulting vector from the cross product is:

   $$(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$$

Interpretation:

The result $(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$ is a vector that lies in the plane defined by $\mathbf{b}$ and $\mathbf{c}$. It reflects how the vector $\mathbf{a}$ influences the plane spanned by $\mathbf{b}$ and $\mathbf{c}$.

Scalar Triple Product

Definition:

The scalar triple product is defined as:

$$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$$

Interpretation:

The scalar triple product gives the volume of the parallelepiped formed by the vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. The volume is the absolute value of the scalar triple product, and its sign indicates the orientation of the vectors.

Proof:

1. Express Using Determinants:

   The scalar triple product can be written as the determinant of a matrix whose rows are the vectors:

   $$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\mid \begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\mid$$

2. Cyclic Permutations:

   The scalar triple product is invariant under cyclic permutations:

   $$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})$$

   This follows from the properties of determinants and the cross product.

Example:

Let $\mathbf{a} = (1, 2, 3)$, $\mathbf{b} = (4, 5, 6)$, and $\mathbf{c} = (7, 8, 9)$.

Scalar Triple Product:

$$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\mid \begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\mid$$

Calculate the determinant:

$$\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix} = 1\cdot (5\cdot9 - 6\cdot8) - 2\cdot (4\cdot9 - 6\cdot7) + 3\cdot (4\cdot8 - 5\cdot7)$$$$= 1\cdot (-3) - 2\cdot (-6) + 3\cdot (-3) = -3 + 12 - 9 = 0$$

Since the result is 0, the vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are coplanar, meaning they lie in the same plane and do not form a three-dimensional parallelepiped.

Summary

• Vector Triple Product: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$. This product gives a vector that can be interpreted as a combination of $\mathbf{b}$ and $\mathbf{c}$, influenced by $\mathbf{a}$.

• Scalar Triple Product: $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$. This product gives a scalar that represents the volume of the parallelepiped formed by $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$, and its sign indicates the orientation.