Question

Prove the following identities. Assume that $\mathbf{u},\mathbf{v},\mathbf{w}$ and x are nonzero vectors in ${\mathbb{R}}^3$. $$(\mathbf{u}\times \mathbf{v})\cdot (\mathbf{w}\times \mathbf{x})=(\mathbf{u}\cdot \mathbf{w})(\mathbf{v}\cdot \mathbf{x})-(\mathbf{u}\cdot \mathbf{x})(\mathbf{v}\cdot \mathbf{w})$$

Short answer

Question: Prove the identity $(\mathbf{u} \times \mathbf{v}) \cdot (\mathbf{w} \times \mathbf{x}) = (\mathbf{u} \cdot \mathbf{w})(\mathbf{v} \cdot \mathbf{x}) - (\mathbf{u} \cdot \mathbf{x})(\mathbf{v} \cdot \mathbf{w})$ for any four vectors $\mathbf{u}, \mathbf{v}, \mathbf{w},$ and $\mathbf{x}$. Answer: The identity can be proved by using the scalar triple product formula and its properties, along with the distributive property of the scalar triple product. By simplifying the expression, we can show that the left-hand side of the identity is equal to the right-hand side.

Step-by-step solution

Write down the scalar triple product formula

Using the scalar triple product, which states that for any three vectors $\mathbf{a},\mathbf{b},\mathbf{c}$: $$[\mathbf{a},\mathbf{b},\mathbf{c}]=[\mathbf{b},\mathbf{c},\mathbf{a}]=[\mathbf{c},\mathbf{a},\mathbf{b}]=-[\mathbf{a},\mathbf{c},\mathbf{b}]$$ we can rewrite LHS as: $$[\mathbf{u},\mathbf{v},(\mathbf{w}\times \mathbf{x})]$$

Use the distributive property of the scalar triple product

Now we apply the distributive property of the scalar triple product, giving us: $$[\mathbf{u},\mathbf{v},(\mathbf{w}\times \mathbf{x})]=(\mathbf{w}\cdot \mathbf{x})[\mathbf{u},\mathbf{v},\mathbf{w}]-(\mathbf{w}\cdot \mathbf{v})[\mathbf{u},\mathbf{v},\mathbf{x}]$$

Rearrange terms using scalar triple product properties

Using the properties of the scalar triple product as mentioned in step 1, we further simplify the expression: $$(\mathbf{w}\cdot \mathbf{x})[\mathbf{u},\mathbf{v},\mathbf{w}]-(\mathbf{w}\cdot \mathbf{v})[\mathbf{u},\mathbf{v},\mathbf{x}]=(\mathbf{u}\cdot \mathbf{w})(\mathbf{v}\cdot \mathbf{x})-(\mathbf{u}\cdot \mathbf{x})(\mathbf{v}\cdot \mathbf{w})$$ Now we see that LHS is equal to RHS, as required: $$(\mathbf{u}\times \mathbf{v})\cdot (\mathbf{w}\times \mathbf{x})=(\mathbf{u}\cdot \mathbf{w})(\mathbf{v}\cdot \mathbf{x})-(\mathbf{u}\cdot \mathbf{x})(\mathbf{v}\cdot \mathbf{w})$$ Thus, the given identity is proved.

Key concepts

Cross Product

The cross product is an operation for vectors in three-dimensional space. It combines two vectors to create a third vector that is perpendicular to both original vectors. The cross product is denoted by the symbol $\times$ and calculated by:

• The direction of the resulting vector follows the right-hand rule. When you point your index finger in the direction of the first vector and your middle finger in the direction of the second vector, your thumb points to the direction of the cross product vector.
• The magnitude of the cross product vector is given by $|\mathbf{a}\times \mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin (\theta )$, where $\theta$ is the angle between vector $\mathbf{a}$ and vector $\mathbf{b}$.
• The cross product is only defined in three dimensions.

It is often used in physics and engineering, especially to determine torque or rotational effects produced by forces.

Scalar Triple Product

The scalar triple product combines three vectors to result in a scalar quantity, which is a measure of the volume of the parallelepiped spanned by the vectors. This process involves both dot and cross products in the form $[\mathbf{a},\mathbf{b},\mathbf{c}]=\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$.

• The order of the vectors matters. If you rotate the order, the sign of the scalar triple product may change. However, the volume remains the same.
• It evaluates to zero if the three vectors are coplanar, indicating that they do not form a parallelepiped with any volume.
• As seen in the exercise, properties of the scalar triple product can simplify vector identities.

Understanding the scalar triple product helps in solving problems involving vector calculus and geometry, like determining collinearity of vectors or finding areas and volumes.

Dot Product

The dot product takes two vectors and produces a scalar. It is a measure of how much one vector extends in the direction of another. The dot product is calculated as:

• $\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos (\theta )$, where $\theta$ is the angle between the two vectors.
• If the dot product is zero, the vectors are orthogonal (perpendicular to each other).
• It is commutative, meaning $\mathbf{a}\cdot \mathbf{b}=\mathbf{b}\cdot \mathbf{a}$.

The dot product is fundamental in physics, especially when computing work done by a force, as work is defined as the dot product of force and displacement vectors.