Problem 19 - Precalculus Mathematics for Calculus

Question

Express the given vector in terms of the unit vectors i, \(\mathbf{j}\). and \(\mathbf{k}\). $$(12,0,2)$$

Step-by-Step Solution

Answer

The vector is expressed as \(12\mathbf{i} + 2\mathbf{k}\).

Step 1: Identify Components

The vector given is \((12, 0, 2)\). These components represent the vector's position in 3D space. The numbers 12, 0, and 2 correspond to the components along the x, y, and z axes, respectively.

Step 2: Understand Vector Notation with Unit Vectors

In vector notation, any vector in 3D space can be expressed using unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) which represent the directions of the x, y, and z axes.

Step 3: Write the Vector in Terms of Unit Vectors

To express the vector \((12, 0, 2)\) in terms of unit vectors, match the components along each axis with the corresponding unit vector: \(12\) matches with \(\mathbf{i}\), \(0\) with \(\mathbf{j}\), and \(2\) with \(\mathbf{k}\).

Step 4: Construct the Vector Expression

Combine the components using the unit vectors to form the expression: \(12\mathbf{i} + 0\mathbf{j} + 2\mathbf{k}\).

Step 5: Simplify the Expression

Since any term multiplied by 0 equals 0, the \(0\mathbf{j}\) term can be omitted; thus, the expression simplifies to \(12\mathbf{i} + 2\mathbf{k}\).

Key Concepts

Unit Vectors

Unit vectors are a fundamental concept in vector mathematics. These vectors serve as the building blocks for more complex vectors and provide a standard way of expressing direction in various dimensions.
In a three-dimensional (3D) space, unit vectors are represented as \(\mathbf{i}, \mathbf{j}, \mathbf{k}\).

• \(\mathbf{i}\) is the unit vector in the direction of the x-axis.
• \(\mathbf{j}\) is the unit vector in the direction of the y-axis.
• \(\mathbf{k}\) is the unit vector in the direction of the z-axis.

Unit vectors have a magnitude of 1. This means they are used purely to show direction without implying any particular length. In vector notation, multiplying these unit vectors by scalar quantities allows us to express any vector as a combination of these three directional components.

3D Space

3D space refers to a coordinate system where any point can be described using three numbers, each representing a different dimension. These dimensions are usually labeled as x, y, and z.
Visualizing a position in 3D space often involves imagining a point on an infinite grid where each axis intersects perpendicularly.

• The x-axis typically represents horizontal position.
• The y-axis typically represents vertical position.
• The z-axis adds depth, extending the concept beyond flat surfaces into a spatial reality.

When dealing with vectors in 3D space, the goal is to specify exactly where a point lies relative to the origin of the coordinate system. This creates a more complete representation of space that includes depth, making it essential in fields such as physics and engineering.

Vector Components

Vector components are the individual parts that make up a vector, each corresponding to one of the coordinate axes. A given vector in 3D space can be broken down into its components along these axes: x, y, and z.
The vector \((12, 0, 2)\) shows how it is positioned in 3D space.

• The number 12 represents movement in the direction of the x-axis.
• The number 0 shows no movement in the y-axis direction.
• The number 2 indicates movement along the z-axis.

Understanding these components allows us to describe vectors using unit vectors. This is done by pairing each component with its respective unit vector, creating an expression like \(12\mathbf{i} + 0\mathbf{j} + 2\mathbf{k}\). However, we often simplify this to \(12\mathbf{i} + 2\mathbf{k}\) because any term with a component of zero can be omitted from the expression.