Problem 53 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Prove the following for vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}: \quad \mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\) \(\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}\)

Step-by-Step Solution

Answer

The Distributive property of Dot Product for vectors has been proven, i.e. \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\)

Step 1: Understanding of Dot Product

The dot product of two vectors is a scalar quantity that is equal to the sum of pair-wise products of the components of vectors. For instances, for two vectors \(\mathbf{u} = [u_1, u_2, ..., u_n]\) and \(\mathbf{v} = [v_1, v_2, ..., v_n]\), the dot product \(\mathbf{u} \cdot \mathbf{v}\) is equal to \(\sum_{i=1}^{n} u_i * v_i\).

Step 2: Expanding the Left Side

Now we have the left side of the equation which is \(\mathbf{u} \cdot (\mathbf{v}+\mathbf{w})\). To simplify this expression, we first perform the addition inside the parenthesis. When we add two vectors, we just add corresponding parts of these vectors. Then we take the dot product of \(\mathbf{u}\) with the resulting vector.

Step 3: Simplifying the Expression

The dot product \(\mathbf{u} \cdot (\mathbf{v}+\mathbf{w})\) simplifies to \(\sum_{i=1}^{n} u_i * (v_i + w_i) = \sum_{i=1}^{n} u_i * v_i + \sum_{i=1}^{n} u_i * w_i\). This is by the distributive property of real numbers.

Step 4: Rewriting into Dot Products

The results of step 3 can be written as dot products. The equation turns into \(\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\).

Step 5: Conclusion

Based on the above steps and the properties of dot product, scalar multiplication and vector addition, we have proven this vector identity: \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\)

Key Concepts

Dot Product

The dot product is a fundamental operation in vector arithmetic. It takes two vectors and combines them to produce a single scalar value.
It is essential in various mathematical and physical applications, like measuring how much one vector extends in the direction of another.

• For vectors \( \mathbf{u} = [u_1, u_2, ..., u_n] \) and \( \mathbf{v} = [v_1, v_2, ..., v_n] \), the dot product is calculated as \( \mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^{n} u_i \cdot v_i \).
• The result is a sum of the product of their corresponding components.
• The dot product is commutative: \( \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u} \).

Each product in the sum belongs to a particular pair of coordinates from the vectors. This sum gives a scalar, reflecting how vectors interact in a shared direction or plane.

Distributive Property

The distributive property in the context of vectors and dot products is similar to what you may know from basic algebra, but applied to vectors.
This property states that the dot product operation distributes over vector addition.

• For vectors, this can be expressed as \( \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \).
• The operation shows how the dot product can "pass through" addition, maintaining its scalar nature.

Essentially, the dot product combines \( \mathbf{u} \) with each of the added vectors separately.
By distributing the dot product individually, each collection of products adds up separately but maintains an overall scalar consistency, aiding in solving equations and in physics.

Vector Addition

Vector addition is a straightforward operation where you simply combine corresponding components of two vectors.
This is foundational to understanding how multiple vectors interact together, creating new directions and magnitudes.

• For vectors \( \mathbf{v} = [v_1, v_2, ..., v_n] \) and \( \mathbf{w} = [w_1, w_2, ..., w_n] \), their addition is calculated as \( \mathbf{v} + \mathbf{w} = [v_1 + w_1, v_2 + w_2, ..., v_n + w_n] \).
• The result is a new vector that geometrically represents a new path or line in the space defined by the vectors.
• This operation is used repeatedly in mathematical proofs and computations, like in solving systems of vectors or determining resultants.

Whether on a plane or in space, vector addition allows the accurate combination and manipulation of directions and magnitudes needed in various fields of study.