Problem 23 - Precalculus Mathematics for Calculus

Question

Express the given vector in terms of the unit vectors i, \(\mathbf{j}\). and \(\mathbf{k}\). $$\mathbf{u}=\langle 0,-2,1\rangle, \quad \mathbf{v}=\langle 1,-1,0\rangle$$

Step-by-Step Solution

Answer

\( \mathbf{u} = -2\mathbf{j} + \mathbf{k} \,;\, \mathbf{v} = \mathbf{i} - \mathbf{j} \).

Step 1: Understanding Vectors and Unit Vectors

Vectors are often represented using the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) which correspond to the x, y, and z components, respectively. The goal is to express any vector \( \mathbf{a} = \langle a_x, a_y, a_z \rangle \) in this form: \( a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} \).

Step 2: Express Vector \( \mathbf{u} \) in Terms of \( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \)

The vector \( \mathbf{u} = \langle 0, -2, 1 \rangle \) has components 0, -2, and 1. Therefore, express \( \mathbf{u} \) as \( 0\mathbf{i} - 2\mathbf{j} + 1\mathbf{k} \).

Step 3: Express Vector \( \mathbf{v} \) in Terms of \( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \)

The vector \( \mathbf{v} = \langle 1, -1, 0 \rangle \) has components 1, -1, and 0. Thus, express \( \mathbf{v} \) as \( 1\mathbf{i} - 1\mathbf{j} + 0\mathbf{k} \).

Key Concepts

Understanding Unit Vectors

Unit vectors are the building blocks of vectors in a coordinate system. Each unit vector has a magnitude of 1 and points in the direction of one of the coordinate axes.
These unit vectors are:

• \( \mathbf{i} \) which points along the x-axis.
• \( \mathbf{j} \) which is along the y-axis.
• \( \mathbf{k} \) that stretches in the z-axis direction.

By using these unit vectors, any vector in a 3D space can be easily expressed in terms of its components along these axes.
For instance, a vector \( \mathbf{a} = \langle a_x, a_y, a_z \rangle \) can be broken down using the unit vectors like this: \( a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} \). This representation shows clearly how much of the vector's magnitude extends in each of the x, y, and z directions.

Exploring Vector Components

Vectors are quantities that have both magnitude and direction. Breaking a vector down into its components along the coordinate axes helps in analyzing and working with the vector in mathematical computations.
Each component of a vector shows the influence of the vector in one dimension.

• The x-component (\( a_x \)) indicates how much the vector extends along the x-axis.
• The y-component (\( a_y \)) reflects the vector's reach along the y-axis.
• The z-component (\( a_z \)) represents the vector's extent along the z-axis.

Understanding these components makes it easier to visualize how the vector acts in a 3D space. For example, in the vector \( \mathbf{u} = \langle 0, -2, 1 \rangle \), it means the vector does not extend in the x-axis, stretches -2 units in the direction of \( \mathbf{j} \) (y-axis), and 1 unit along the z-axis.

Navigating the 3D Coordinate System

In 3D coordinate systems, we use three axes – the x-axis, y-axis, and z-axis – to determine the position of any point or object in space. These axes meet at a central point called the origin. Each point in this space can be expressed as a vector composed of three components corresponding to these axes.

• An important feature of the 3D coordinate system is its ability to give depth (z-axis) along with length (x-axis) and height (y-axis).
• It is useful not just in physics and engineering, but also in fields like computer graphics and astronomy.

When solving problems involving vectors, engaging with the 3D coordinate system allows for clear visualization and mapping of vectors in a three-dimensional space. It greatly simplifies the task of adding vectors, finding their resultant, and extracting information like magnitude and direction. By understanding and utilizing the 3D coordinate system, problem-solving becomes more intuitive and manageable.