Problem 21 - Precalculus Mathematics for Calculus

Question

Express the given vector in terms of the unit vectors i, \(\mathbf{j}\). and \(\mathbf{k}\). $$(3,-3,0)$$

Step-by-Step Solution

Answer

The vector is expressed as \(3\mathbf{i} - 3\mathbf{j} + 0\mathbf{k}\).

Step 1: Understanding the Vector Notation

Given a vector \((3, -3, 0)\), we start by recognizing each component in terms of its respective axis in a 3D coordinate system. In this system, the first number is the x-component, the second is the y-component, and the third is the z-component.

Step 2: Expressing in Unit Vectors

Unit vectors are generally represented by \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), which are the unit vectors along the x, y, and z axes, respectively. To express the vector, multiply each component with its respective unit vector.

Step 3: Writing the Vector in Unit Vector Form

The vector \((3, -3, 0)\) can be expressed in terms of the unit vectors as follows: \[3\mathbf{i} - 3\mathbf{j} + 0\mathbf{k}.\]This is the expression in terms of the unit vectors.

Key Concepts

Unit Vectors

Unit vectors play a crucial role in vector notation as they provide a way to describe directions in any given space. These vectors have a magnitude of one and specify direction only.

• The unit vector in the direction of the x-axis is denoted as \(\mathbf{i}\), and it equals \((1, 0, 0)\).
• The unit vector for the y-axis is represented by \(\mathbf{j}\), corresponding to \((0, 1, 0)\).
• For the z-axis, the unit vector is \(\mathbf{k}\), expressed as \((0, 0, 1)\).

By utilizing unit vectors, you can easily construct any vector within a coordinate system. To represent a vector like \((3, -3, 0)\) in terms of unit vectors, each component's magnitude is assigned to its respective unit vector. Hence, it simplifies vector expressions by aligning them according to each axis.

3D Coordinate System

The 3D coordinate system provides a method to describe the position of any point in three-dimensional space using three coordinates. It is an extension of the 2D coordinate system, adding the z-axis to the x and y axes.
Each point in this system is represented as \((x, y, z)\), where:

• \(x\) is the value on the horizontal axis (x-axis).
• \(y\) is the value on the vertical axis (y-axis).
• \(z\) represents the depth, adding a third level to the plane in what we refer to as 3D space.

When working in 3D, it's crucial to note that vectors like \((3, -3, 0)\) can occupy any part of this space by modifying their components. The addition of the z-axis offers a new dimension of possibilities, facilitating the representation of objects and directions that we encounter in the real world.

Vector Components

Vector components are the building blocks of vectors in any dimensional space. They break down a vector into parts that align with the respective axes of the coordinate system.
For a vector \((x, y, z)\) in 3D space, each value inside the parentheses represents the component of the vector along a particular axis:

• The **x-component** determines the vector's extent along the x-axis.
• The **y-component** specifies the vector's reach along the y-axis.
• The **z-component** indicates how far the vector stretches along the z-axis.

By understanding vector components, you enable precise manipulation and combination of vectors, essential for physics, engineering, and various fields where spatial understanding is key. The vector \((3, -3, 0)\) can thus be decomposed into its components along the respective axes and expressed through unit vectors as \(3\mathbf{i} - 3\mathbf{j} + 0\mathbf{k}\). This way, it becomes easier to analyze and visualize complex vector calculations.