Problem 30 - Precalculus Mathematics for Calculus

Question

Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\). $$\mathbf{u}=\langle 0,-5\rangle$$

Step-by-Step Solution

Answer

\( \mathbf{u} = -5\mathbf{j} \).

Step 1: Understand the Components of the Vector

The vector \( \mathbf{u} = \langle 0, -5 \rangle \) is given in component form. This means it has two parts: the first part represents the \(x\)-component, and the second part represents the \(y\)-component of the vector. Here, the \(x\)-component is 0 and the \(y\)-component is -5.

Step 2: Express the Vector in Terms of \(\mathbf{i}\) and \(\mathbf{j}\)

Each vector can be expressed in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), which represent the x- and y-directions, respectively. Thus, \( \langle a, b \rangle \) can be written as \( a \mathbf{i} + b \mathbf{j} \). For the vector \( \mathbf{u} = \langle 0, -5 \rangle \), this becomes \( 0\mathbf{i} + (-5)\mathbf{j} \).

Step 3: Simplify the Expression

Simplify the expression by removing any terms with a zero coefficient. Since \( 0\mathbf{i} \) has a zero coefficient, it can be omitted from the expression. The vector is simplified to \( -5\mathbf{j} \).

Key Concepts

Unit Vectors

Unit vectors are fundamental in vector mathematics since they serve as baseline building blocks for constructing other vectors. A unit vector has a magnitude of exactly one and serves to point in a specific direction.

• In a two-dimensional space, the standard unit vectors are \(\mathbf{i}\) and \(\mathbf{j}\).
• \(\mathbf{i}\) is the unit vector in the x-direction, represented as \(\langle 1, 0 \rangle\).
• \(\mathbf{j}\) is the unit vector in the y-direction, represented as \(\langle 0, 1 \rangle\).

Knowing and understanding unit vectors is indispensable for working with vector components, which uses unit vectors as reference points.

Vector Components

Vector components break down a vector into perpendicular directions based on the Cartesian coordinate system. This decomposition makes it easier to analyze and work with vectors.

• For any vector \(\mathbf{v} = \langle a, b \rangle\),
  "a" is the x-component of the vector.
• "b" is the y-component of the vector.

By understanding vector components, we can reconstruct vectors using unit vectors and achieve a deeper understanding of how vectors interact in space.
Apportioning a vector into these components allows us to solve complex problems by tackling simple linear equations.
In our example, the vector \(\mathbf{u} = \langle 0, -5 \rangle\) has 0 as the x-component and -5 as the y-component.

Linear Combinations

A linear combination is critical in expressing one vector using a set of vectors as a basis. This involves scaling the basis vectors and adding them to find the desired result.

• In our context, vectors like \(\mathbf{u} = \langle 0, -5 \rangle\) can be depicted using unit vectors \(\mathbf{i}\) and \(\mathbf{j}\).
• The linear combination is expressed as \(a\mathbf{i} + b\mathbf{j}\), where a and b are coefficients.
• This allows the representation of any vector in terms of basic directions and simplifies vector operations.

By using linear combinations, vectors can be effectively manipulated and understood within their environments, providing a solid framework for both theoretical explorations and practical applications.

Vector Notation

Vector notation is a shorthand method of representing vectors using symbols and is crucial for efficient communication of mathematical ideas.
There are several ways to denote vectors:

• Component form, shown as \(\langle x, y \rangle\), specifies the x and y components directly.
• The unit vector form, like \(a\mathbf{i} + b\mathbf{j}\), uses unit vectors to indicate directionality.
• In written text, vectors are often bolded or accented with overhead arrows, as in \(\mathbf{v}\) or \(\vec{v}\).

This variety in vector notation allows flexibility in solving problems and makes it easier to convey complex vector information with simplicity and precision.
Regardless of the context, vector notation helps convey complex concepts succinctly and effectively, aiding in both teaching and communication.