Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\). $$\mathbf{v}=-\mathbf{u}+\mathbf{w}$$

Step-by-Step Solution

Answer

The component form of the vector \(v\) is (-1, 3). After sketching, the vectors \(u\), \(w\), \(-u\), and \(v\) will have origins at the point (0, 0) and heads at the points (2, -1), (1, 2), (-2, 1), and (-1, 3) respectively.

Step 1: Compute the Negative of Vector \(u\)

To compute the negative of a vector, we simply change the sign of each of its components. Hence, the negative of vector \(u = 2i - j\) will be \(-u = -2i + j\).

Step 2: Compute the Sum of Vectors \(-u\) and \(w\)

To add two vectors, we add their corresponding components. Thus, the sum \(-u + w = (-2i + j) + (i + 2j) = -i + 3j\). So, in component form, vector \(v\) is (-1, 3).

Step 3: Sketching the Vectors

Vector \(u\) is sketched from the origin to the point (2, -1), vector \(w\) to the point (1, 2), and the resulting vector \(v\) from the vector operation ends at the point (-1, 3). The negative of vector \(u\) is just vector \(u\) reflected across the origin to (-2, 1). Adding \(w\) to \(-u\) is equivalent to placing the tail of \(w\) at the head of \(-u\) and drawing the resultant vector from the tail of \(-u\) to the head of \(w\) which ends at (-1, 3), the coordinates of \(v\).

Key Concepts

Component Form

When dealing with vectors in a plane, the component form helps in breaking down a vector into its horizontal and vertical components. Think of it like pinpointing the exact coordinates on a grid. Each vector can be expressed using these components, often denoted by unit vectors like \(\mathbf{i}\) and \(\mathbf{j}\), representing the horizontal and vertical directions, respectively.
The component form of a vector is written as \( (a, b) \), where \( a \) and \( b \) are numerical values indicating how far the vector moves horizontally and vertically from its origin. Using the example above, vector \( \mathbf{u} = 2 \mathbf{i} - \mathbf{j} \) means it has a horizontal component of 2 and a vertical component of -1. Hence, it's written as \( (2, -1) \) in component form.

• Comprehensive understanding of each component is crucial when performing vector operations such as addition or subtraction.
• This allows for transformations like scaling and direction changes to be calculated with simplicity.

Vector Representation

Vectors can be represented in different ways, such as by using graphics or mathematical expressions. Graphically, vectors are depicted as arrows with direction and magnitude. The direction is where the vector points, and the magnitude is its length. Mathematically, vectors are often expressed in terms of their components, making calculations straightforward.
In computational settings, unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) serve as the basic representations in 2D space. By scaling these unit vectors with numerical coefficients, any vector on the plane can be described uniquely through addition and subtraction.
For example, our original vectors \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} + 2\mathbf{j} \) are represented by how much of each unit vector they comprise. Vector \( \mathbf{u} \) means moving 2 units along \( \mathbf{i} \) and moving down 1 unit along \( \mathbf{j} \). In drawing these vectors on a grid, each corresponds to motion from the origin to a specific endpoint, creating visible geometric shapes like triangles or parallelograms when combined.

Vector Operations

Operating with vectors involves actions like addition, subtraction, and multiplication by a scalar, each changing the vector's characteristics. These operations adhere to specific rules that simplify working with vectors in various calculations.
1. **Addition**: To add vectors, simply sum their respective components. This process determines a new vector that effectively "li” arranges two individual vectors into one continuous line.
2. **Subtraction**: Subtraction involves subtracting corresponding components. Geometrically, it represents the vector that "connects" the heads of the subtracted vector from the tail of the original vector.
3. **Multiplication by a Scalar**: Multiplying a vector by a scalar alters its magnitude but not its direction unless the scalar is negative, which also reflects it.
In our exercise, the operation \(-\mathbf{u} + \mathbf{w} = \mathbf{v}\) is handled stepwise:

• First, compute \(-\mathbf{u}\) by negating each component of \( \mathbf{u} \).
• Then, add \( \mathbf{w} \)'s components, resulting in a new vector representation.

Each operation modifies the original vectors' path, affecting their end location and ultimately depicting a combined effect visually and numerically.