Question

Prove the distributive property: $$ \mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w} $$

Short answer

Using the component form and applying the distributive property of multiplication, we can prove \( \textbf{u} \bullet (\textbf{v} + \textbf{w}) = \textbf{u} \bullet \textbf{v} + \textbf{u} \bullet \textbf{w} \).

Step-by-step solution

Understand the Distributive Property

The distributive property in vector algebra states that the dot product of a vector \( \textbf{u} \) with the sum of two vectors \( \textbf{v} \) and \( \textbf{w} \) is equal to the sum of the dot product of \( \textbf{u} \) with \( \textbf{v} \) and the dot product of \( \textbf{u} \) with \( \textbf{w} \). Mathematically, it is represented as \( \textbf{u} \bullet (\textbf{v} + \textbf{w}) = \textbf{u} \bullet \textbf{v} + \textbf{u} \bullet \textbf{w} \).

Express Vectors in Component Form

Let \( \textbf{u} = (u_1, u_2, u_3) \), \( \textbf{v} = (v_1, v_2, v_3) \), and \( \textbf{w} = (w_1, w_2, w_3) \). These are vectors in 3-dimensional space with components in the \( x \), \( y \), and \( z \) directions.

Apply the Dot Product Definition

Recall that the dot product of two vectors \( \textbf{a} = (a_1, a_2, a_3) \) and \( \textbf{b} = (b_1, b_2, b_3) \) is given by: \[ \textbf{a} \bullet \textbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

Distributive Property Using Components

Using the component forms, we need to prove: \[ \textbf{u} \bullet (\textbf{v} + \textbf{w}) = (u_1, u_2, u_3) \bullet ((v_1, v_2, v_3) + (w_1, w_2, w_3)) \] Since vector addition is done component-wise, we get: \[ (v_1 + w_1, v_2 + w_2, v_3 + w_3) \]

Compute the Dot Product

Now, compute the dot product of \( \textbf{u} \) with \( \textbf{v} + \textbf{w} \): \[ (u_1, u_2, u_3) \bullet (v_1 + w_1, v_2 + w_2, v_3 + w_3) = u_1(v_1 + w_1) + u_2(v_2 + w_2) + u_3(v_3 + w_3) \]

Apply Distributive Property of Multiplication

Expand the right-hand side using the distributive property of multiplication: \[ u_1v_1 + u_1w_1 + u_2v_2 + u_2w_2 + u_3v_3 + u_3w_3 \]

Group Like Terms

Re-arrange the terms to group the dot products separately: \[ (u_1v_1 + u_2v_2 + u_3v_3) + (u_1w_1 + u_2w_2 + u_3w_3) \]

Use Dot Product Definition

Recognize that the grouped terms are dot products: \[ (u_1, u_2, u_3) \bullet (v_1, v_2, v_3) + (u_1, u_2, u_3) \bullet (w_1, w_2, w_3) = \textbf{u} \bullet \textbf{v} + \textbf{u} \bullet \textbf{w} \]

Conclude the Proof

Therefore, we have proven that \[ \textbf{u} \bullet (\textbf{v} + \textbf{w}) = \textbf{u} \bullet \textbf{v} + \textbf{u} \bullet \textbf{w} \]

Key concepts

Vector Algebra

Vector algebra is a branch of mathematics that deals with vectors and operations on vectors. A vector is a quantity that has both magnitude (or length) and direction. Vectors are usually represented by arrows, where the length of the arrow represents the magnitude, and the direction of the arrow represents the direction of the vector. For example, in a 2-dimensional plane, a vector can be represented as \( \mathbf{v} = (v_1, v_2) \).

Operations on vectors include addition, subtraction, and scalar multiplication. Vector addition is performed component-wise, meaning that the corresponding components of the vectors are added together. Scalar multiplication involves multiplying each component of the vector by a scalar (a real number).

Understanding vector algebra is essential for working with concepts like dot product and the distributive property within vector spaces.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It takes two vectors and returns a scalar. The dot product of two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \) is defined as:
\[ \mathbf{a} \bullet \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
The result is a single number, not a vector.

The dot product has several important properties:

• It measures the projection of one vector onto another.
• It is commutative, meaning \( \mathbf{a} \bullet \mathbf{b} = \mathbf{b} \bullet \mathbf{a} \).
• If the dot product is zero, the vectors are perpendicular.

By applying the dot product to various problems, we can understand the relationship between vectors in different dimensions.

Vector Components

Vectors can be broken down into components, which represent the influence of the vector in each axis or direction. For example, in 3-dimensional space, a vector \( \mathbf{u} = (u_1, u_2, u_3) \) has components in the \( x \)-, \( y \)-, and \( z \)-directions. These components are essential in performing operations like vector addition and the dot product.

When we add two vectors, we add their corresponding components:
\[ \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3) \]

Similarly, when we calculate the dot product, we multiply the corresponding components and then sum the results:
\[ \mathbf{u} \bullet \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \]

Understanding vector components helps in visualizing and performing various vector operations, making math problems much simpler to solve.

Distributive Property

The distributive property is an essential concept in vector algebra. It states that the dot product of a vector \( \mathbf{u} \) with the sum of two vectors \( \mathbf{v} \) and \( \mathbf{w} \) is equal to the sum of the dot product of \( \mathbf{u} \) with \( \mathbf{v} \) and the dot product of \( \mathbf{u} \) with \( \mathbf{w} \). Mathematically, this is written as:
\[ \mathbf{u} \bullet (\mathbf{v} + \mathbf{w}) = \mathbf{u} \bullet \mathbf{v} + \mathbf{u} \bullet \mathbf{w} \]

To prove this property, express the vectors in component form. Let \( \mathbf{u} = (u_1, u_2, u_3) \), \( \mathbf{v} = (v_1, v_2, v_3) \), and \( \mathbf{w} = (w_1, w_2, w_3) \). The addition of vectors \( \mathbf{v} \) and \( \mathbf{w} \) can be written as:
\[ (v_1, v_2, v_3) + (w_1, w_2, w_3) = (v_1 + w_1, v_2 + w_2, v_3 + w_3) \]

Now, compute the dot product:
\[ \mathbf{u} \bullet (\mathbf{v} + \mathbf{w}) = (u_1, u_2, u_3) \bullet (v_1 + w_1, v_2 + w_2, v_3 + w_3) = u_1(v_1 + w_1) + u_2(v_2 + w_2) + u_3(v_3 + w_3) \]
Next, apply the distributive property of multiplication to expand the terms:
\[ u_1v_1 + u_1w_1 + u_2v_2 + u_2w_2 + u_3v_3 + u_3w_3 \]
Group like terms:
\[ (u_1v_1 + u_2v_2 + u_3v_3) + (u_1w_1 + u_2w_2 + u_3w_3) \]
Recognize that these are the dot products of \( \mathbf{u} \) with \( \mathbf{v} \) and \( \mathbf{u} \) with \( \mathbf{w} \):
\[ \mathbf{u} \bullet \mathbf{v} + \mathbf{u} \bullet \mathbf{w} \]
Thus, the distributive property is proved.

Proof in Algebra

Proofs in algebra are logical arguments that validate the truth of a mathematical statement. In the context of vector algebra, proofs often involve showing that specific properties hold under the defined operations of vectors. The distributive property of the dot product, for example, is a statement that we can prove using algebraic manipulation of vector components.

The proof involves breaking down vectors into their components and applying known operations like vector addition and multiplication. By following step-by-step reasoning, we transform and simplify the expressions to arrive at the desired result.

This methodical process ensures that the proposed property or theorem is true in all cases. Understanding proofs in algebra helps build a solid foundation in mathematical reasoning and problem-solving.