Problem 81 - Algebra and Trigonometry

Question

Use the vectors $$\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad \text { and } \quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j}$$ to prove the given property. $$ \mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w} $$

Step-by-Step Solution

Answer

The given vector property \( \mathbf{u} \cdot(\mathbf{v}+\mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \) holds true due to the distributive property of the dot product over vector addition.

Step 1: Specify the vectors

The provided vectors are \(\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}\), \(\mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}\), and \(\mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j}\).

Step 2: Compute the dot product

The dot product of two vectors involves multiplying the corresponding components of the vectors and adding these products. \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})= a_{1}(a_{2} + a_{3}) + b_{1}(b_{2} +b_{3}) = (a_{1}a_{2} + a_{1}a_{3}) + (b_{1}b_{2} + b_{1}b_{3})\). This is the dot product of \(\mathbf{u}\) with the sum of \(\mathbf{v}\) and \(\mathbf{w}\).

Step 3: Distribute and simplify

Next, distribute the dot product formula over the sum of the vectors. This results in \( \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} = (a_{1}a_{2} + b_{1}b_{2}) + (a_{1}a_{3} + b_{1}b_{3})\). By the distributive property of multiplication over addition, this expression simplifies to the expression obtained in Step 2, confirming the equality.

Key Concepts

Understanding the Dot Product

The dot product, also known as the scalar product, is a fundamental concept in vector algebra. It involves multiplying the corresponding components of two vectors and then summing these products to result in a scalar quantity, not another vector. This operation is crucial because it gives insight into the relationship between the two vectors, such as their angle or direction. To calculate the dot product of two-dimensional vectors such as \( \mathbf{u} = a_1 \mathbf{i} + b_1 \mathbf{j} \) and \( \mathbf{v} = a_2 \mathbf{i} + b_2 \mathbf{j} \), the formula is: \[ \mathbf{u} \cdot \mathbf{v} = a_1a_2 + b_1b_2 \] This result is derived from the summation of the products of their respective components. Remember, the dot product captures the magnitude of the projection of one vector onto the other; hence their relative alignment influences the result significantly.

Exploring the Distributive Property in Vector Algebra

The distributive property is a key principle commonly seen in algebra, but it also applies well to vectors in the realm of dot products. Essentially, this property allows for the distribution of operations over addition, ensuring that operations with vectors remain structured and systematic.In vector notation, when dealing with the dot product, the distributive property implies:\[ \mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} \] This formula illustrates that when you have a vector multiplied by the addition of two other vectors, you can separate these operations into individual dot products with the same vector \( \mathbf{u} \). This result is helpful because it simplifies complex vector equations and maintains harmonic relationships across vector operations. Consistently applying the distributive property ensures that calculations are both simplified and correct.

Vector Addition Basics

Vector addition is the process of combining two or more vectors to form a resultant vector. This operation is performed by adding the corresponding components from each vector. Imagine vector addition as moving along one vector, and then continuing along another to reach the final destination. For vectors \( \mathbf{v} = a_2 \mathbf{i} + b_2 \mathbf{j} \) and \( \mathbf{w} = a_3 \mathbf{i} + b_3 \mathbf{j} \), vector addition is calculated as follows:

• Sum of i-components: \( a_2 + a_3 \)
• Sum of j-components: \( b_2 + b_3 \)

The resultant vector will be \( (a_2 + a_3) \mathbf{i} + (b_2 + b_3) \mathbf{j} \). This way of adding vectors maintains the physical intuition of moving in space across two dimensions, summing their influences towards forming one single trajectory or direction.

Decomposing Vectors into Components

The notion of breaking down a vector into its components is foundational in understanding vector algebra. Vectors in two-dimensional space are typically expressed in terms of their horizontal (i-component) and vertical (j-component) components, often denoted in the format \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} \). Each vector component describes how far and in which direction the vector extends in each coordinate direction. This decomposition helps visualize and evaluate the vector's behavior in each principal direction independently. Key characteristics of vector components:

• The i-component indicates the left-right orientation.
• The j-component indicates the up-down orientation.

Understanding vectors in terms of their individual components allows for simpler computations and clear geometric interpretations of vector operations, such as the dot product or vector addition.