Problem 59 - Algebra and Trigonometry

Question

In Exercises \(57-60,\) let $$\begin{aligned}\mathbf{u} &=a_{1} \mathbf{i}+b_{1} \mathbf{j} \\\\\mathbf{v} &=a_{2} \mathbf{i}+b_{2} \mathbf{j} \\\\\mathbf{w} &=a_{3} \mathbf{i}+b_{3} \mathbf{j}\end{aligned} $$ Prove each property by obtaining the vector on each side of the equation. Have you proved a distributive, associative, or commutative property of vectors? $$c(\mathbf{u}+\mathbf{v})=c \mathbf{u}+c \mathbf{v}$$

Step-by-Step Solution

Answer

The property \(c(\mathbf{u}+\mathbf{v}) = c\mathbf{u} + c\mathbf{v}\) has been proven. This is a distributive property of vector operations.

Step 1: Expand the left hand side of the equation

Start by expanding the left hand side (LHS) of the equation using the definition of vector addition. If \(\mathbf{u}\) = \(a_{1} \mathbf{i}+b_{1} \mathbf{j}\) and \(\mathbf{v}\) = \(a_{2} \mathbf{i}+b_{2} \mathbf{j}\), then \(\mathbf{u}+\mathbf{v}\) would be \((a_{1}+a_{2})\mathbf{i} + (b_{1}+b_{2})\mathbf{j}\). Multiplying this by the scalar \(c\) gives us \(c(\mathbf{u}+\mathbf{v}) = c(a_{1}+a_{2})\mathbf{i} + c(b_{1}+b_{2})\mathbf{j}\).

Step 2: Expand the right hand side of the equation

Now, expand the right hand side (RHS) of the equation. This involves a scalar multiplying each of the individual vectors \(\mathbf{u}\) and \(\mathbf{v}\). Therefore, \(c\mathbf{u}+c\mathbf{v} = (c a_{1}\mathbf{i} + c b_{1}\mathbf{j}) + (c a_{2}\mathbf{i} + c b_{2}\mathbf{j}) = (c a_{1}+c a_{2})\mathbf{i}+(c b_{1}+c b_{2})\mathbf{j}\).

Step 3: Compare both sides of the equation

Finally, compare the LHS and the RHS of the equation. As can be seen, in both cases, the result is a vector whose \(\mathbf{i}\) component is the sum of the \(\mathbf{i}\) components of \(c\mathbf{u}\) and \(c\mathbf{v}\), and whose \(\mathbf{j}\) component is the sum of the \(\mathbf{j}\) components of \(c\mathbf{u}\) and \(c\mathbf{v}\). Thus, it proves that indeed, \(c(\mathbf{u}+\mathbf{v}) = c\mathbf{u} + c\mathbf{v}\).

Key Concepts

Vector Addition

Vector addition is a fundamental concept in physics and engineering, as it helps describe movements and forces in a space. For vectors in a two-dimensional plane represented as \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \), adding them involves combining their respective components. The components are the projections of the vector onto the axes, usually labeled \( \mathbf{i} \) for the x-axis and \( \mathbf{j} \) for the y-axis.

Now, when vectors \( \mathbf{u} \) and \( \mathbf{v} \) are added, their corresponding components are added together. Mathematically, this is expressed as \( \mathbf{u} + \mathbf{v} = (a_1 + a_2)\mathbf{i} + (b_1 + b_2)\mathbf{j} \). Here, \( a_1 + a_2 \) is the x-component, and \( b_1 + b_2 \) is the y-component of the resulting vector. The process is intuitive, like adding distances in the same direction, and it complies with the commutative property where \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \).

Understanding vector addition is crucial for solving problems involving simultaneous movements or forces acting in different directions. It allows for a clear representation of how individual vectors combine to form a resultant vector in any direction within the plane.

Scalar Multiplication

When working with vectors, scalar multiplication allows us to stretch or shrink a vector's magnitude while maintaining its direction or reverse the direction if the scalar is negative. It is called 'scalar' because the quantity by which you're multiplying is a single value, not a vector.

In our example, scalar multiplication is demonstrated by multiplying vector \( \mathbf{u} \) by a scalar \( c \), resulting in \( c\mathbf{u} \) which equals \( c a_{1}\mathbf{i} + c b_{1}\mathbf{j} \). Every component of the vector is multiplied by \( c \), effectively scaling the vector's size without changing its orientation unless \( c \) is negative, where the vector flips direction.

This operation is pivotal in physics for understanding concepts like velocity or force, where a change in magnitude is frequently required without altering the direction. Scalar multiplication is distributive over vector addition, as shown in the aforementioned exercise, complying with the equation \( c(\mathbf{u}+\mathbf{v}) = c\mathbf{u} + c\mathbf{v} \). This property is immensely useful in simplifying complex vector expressions.

Vector Components

The idea of vector components is an integral part of understanding vectors as a whole. A vector in a plane can be broken down into parts that run parallel to the x-axis (horizontal) and y-axis (vertical). These parts are called the x-component and the y-component of the vector, usually represented by \( \mathbf{i} \) and \( \mathbf{j} \) components, respectively.

For instance, a vector \( \mathbf{u} \) can be represented in component form as \( a_1\mathbf{i} + b_1\mathbf{j} \), where \( a_1 \) is the magnitude of the x-component and \( b_1 \) is the magnitude of the y-component. This representation provides clarity on how much the vector moves in each direction of the corresponding axes.

Understanding each component is crucial for various applications, from plotting the vector's position to calculating work done against a force. It also aids in performing vector additions and multiplications accurately, as each component is dealt with individually. The step-by-step solution provided demonstrates how understanding the components can simplify proving vector properties like distributive over scalar multiplication.