Problem 28 - Precalculus Mathematics for Calculus

Question

Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\). $$\mathbf{u}=\langle- 2,10\rangle$$

Step-by-Step Solution

Answer

\(\mathbf{u} = -2\mathbf{i} + 10\mathbf{j}\)

Step 1: Understanding the Vector Components

The vector \( \mathbf{u} = \langle -2, 10 \rangle \) indicates the components of the vector in a two-dimensional space. The first component, \( -2 \), is the horizontal component along the \( x \)-axis, and the second component, \( 10 \), is the vertical component along the \( y \)-axis.

Step 2: Expressing the Vector with Unit Vectors

In two-dimensional space, the vector components can be expressed using the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \). The vector \( \mathbf{i} \) represents one unit in the direction of the \( x \)-axis (horizontally), and the vector \( \mathbf{j} \) represents one unit in the direction of the \( y \)-axis (vertically).

Step 3: Combining Components with Unit Vectors

To express \( \mathbf{u} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \), we multiply each component by its corresponding unit vector: \(-2\mathbf{i} + 10\mathbf{j}\). This indicates that the vector has a horizontal component of \(-2\) and a vertical component of \(10\).

Key Concepts

Unit Vectors

Unit vectors are fundamental building blocks in vector mathematics. They are vectors with a magnitude of exactly one, making them perfect for representing direction without scaling the size. In two-dimensional space, we commonly encounter the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\).

• \(\mathbf{i}\) is oriented along the x-axis, signaling one unit in the horizontal direction.
• \(\mathbf{j}\) is oriented along the y-axis, representing one unit vertically.

These unit vectors allow us to construct other vectors by scaling them appropriately. For instance, any vector \(\mathbf{a}\) with horizontal component \(a_1\) and vertical component \(a_2\) can be written as \(a_1\mathbf{i} + a_2\mathbf{j}\). Thus, unit vectors help us express vectors in a form that highlights their directions and respective magnitudes.

Two-Dimensional Space

When discussing vectors like \(\mathbf{u} = \langle -2, 10 \rangle\), we are operating within a two-dimensional space. This space comprises two axes: the horizontal x-axis and the vertical y-axis.

• Vectors in this space are defined by these two coordinates, each representing a point's position relative to the origin.
• The two-dimensional space is visualized as a flat plane, where any point or vector can be positioned using only x and y coordinates.

Understanding this concept is crucial because it allows us to map real-world movements and forces in a simplified environment.
Graphically, each vector's tail begins at the origin, \(\langle 0, 0 \rangle\), and stretches to its point \(\langle x, y \rangle\). This clarity in visualization aids in computations and practical applications across various fields like physics and engineering.

Components of a Vector

Each vector in a two-dimensional space can be broken down into its components, which describe the vector's influence along each axis. The vector \(\mathbf{u} = \langle -2, 10 \rangle\) includes two components:

• The horizontal component is \(-2\), indicating movement in the negative direction on the x-axis.
• The vertical component is \(10\), showing upward movement along the y-axis.

These components essentially express how much the vector "pushes" or "pulls" in each direction. To easily manage and manipulate vectors, we express these components using unit vectors, resulting in the vector formula: \(-2\mathbf{i} + 10\mathbf{j}\). This representation:

• Maintains clarity on how the vector interacts with each dimension individually.
• Fosters accuracy in mathematical operations like vector addition, subtraction, and scalar multiplication.

Grasping vector components simplifies the understanding of vector direction and magnitude.