Question

Simplify \((a \mathbf{u}+b \mathbf{v}) \times(c \mathbf{u}+d \mathbf{v})\).

Short answer

\((ad - bc)(\mathbf{u} \times \mathbf{v})\)

Step-by-step solution

Distribute the Cross Product

The expression given, \((a \mathbf{u}+b \mathbf{v}) \times (c \mathbf{u}+d \mathbf{v})\), can be distributed using the distributive property of cross products as \(a \mathbf{u} \times c \mathbf{u} + a \mathbf{u} \times d \mathbf{v} + b \mathbf{v} \times c \mathbf{u} + b \mathbf{v} \times d \mathbf{v}\).

Simplify Using Cross Product Properties

Any vector crossed with itself is zero, i.e., \( \mathbf{u} \times \mathbf{u} = \mathbf{v} \times \mathbf{v} = 0\). This gives us: \(a \mathbf{u} \times c \mathbf{u} = 0\) and \(b \mathbf{v} \times d \mathbf{v} = 0\). The expression simplifies to \(a \mathbf{u} \times d \mathbf{v} + b \mathbf{v} \times c \mathbf{u}\).

Rearrange Remaining Terms

Since the cross product is anti-commutative, \(\mathbf{v} \times \mathbf{u} = -\mathbf{u} \times \mathbf{v}\), simplify the expression further. Thus, \(b \mathbf{v} \times c \mathbf{u} = -bc (\mathbf{u} \times \mathbf{v})\). The total expression is \(ad(\mathbf{u} \times \mathbf{v}) - bc (\mathbf{u} \times \mathbf{v})\).

Factor the Expression

Factor out \(\mathbf{u} \times \mathbf{v}\) from the expression: \((ad - bc)(\mathbf{u} \times \mathbf{v})\).

Key concepts

Distributive Property in Vector Algebra

The distributive property is a key aspect in algebra that extends to vector algebra as well. In simple terms, for an operation like multiplication, it means you can multiply a sum by multiplying each addend separately, and then add the products. When working with vectors, this property is crucial especially with the cross product.

For the expression \((a \mathbf{u}+b \mathbf{v}) \times (c \mathbf{u}+d \mathbf{v})\), applying the distributive property results in \(a \mathbf{u} \times c \mathbf{u} + a \mathbf{u} \times d \mathbf{v} + b \mathbf{v} \times c \mathbf{u} + b \mathbf{v} \times d \mathbf{v}\). This is because each term from the first vector is crossed with each term from the second vector.

This process might seem complex at first glance, but if you remember how distributive property functions with regular numbers, it gets easier to apply with cross products and vectors. It ensures that every part of the expressions is interacted with, making sure no piece is left uncalculated.

Anti-Commutative Property of Cross Products

The anti-commutative property is another interesting feature of vector cross products. It states that the cross product of two vectors is not commutative, meaning \(\mathbf{a} \times \mathbf{b} eq \mathbf{b} \times \mathbf{a}\). In fact, \(\mathbf{b} \times \mathbf{a} = -\mathbf{a} \times \mathbf{b}\). This means the order of vectors in a cross product changes the direction of the result.

This property is essential in simplifying expressions involving cross products. In the expression from our example, after using the distributive property, we have terms like \(b \mathbf{v} \times c \mathbf{u}\). According to anti-commutative property, this can be rewritten as \(-bc(\mathbf{u} \times \mathbf{v})\). This simplification is crucial as it often leads to easier factorization or further simplification of vector expressions.

Remembering that swapping vectors in a cross product introduces a negative sign helps you avoid mistakes when manipulating vector equations.

Understanding Vector Algebra

Vector algebra involves operations between vectors in a vector space. These operations include addition, scalar multiplication, and notably, cross products. Vectors themselves are quantities having both magnitude and direction, and vector algebra allows complex operations to be carried out in a systematic way.

A fundamental part of vector algebra is understanding how different properties, like distributive and anti-commutative properties, interact with vector operations like cross products and dot products. The cross product \(\mathbf{a} \times \mathbf{b}\) is a vector perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\), with a magnitude proportional to the area of the parallelogram that the vectors span.

In our exercise, understanding vector algebra allows us to simplify the expression \((a \mathbf{u}+b \mathbf{v}) \times(c \mathbf{u}+d \mathbf{v})\) efficiently. By mastering these concepts, one can solve real-world problems ranging from physics to computer graphics that depend on calculating vector fields and forces.