Problem 22 - Algebra and Trigonometry

Question

Express the given vector in terms of the unit vectors i, j, and k. $$ \left\langle- a, \frac{1}{3} a, 4\right\rangle $$

Step-by-Step Solution

Answer

\(-a\mathbf{i} + \frac{1}{3}a\mathbf{j} + 4\mathbf{k}\)

Step 1: Identifying the Components

The vector given is \( \langle -a, \frac{1}{3}a, 4 \rangle \). This vector is in the form of \( \langle x, y, z \rangle \) where \( x = -a \), \( y = \frac{1}{3}a \), and \( z = 4 \).

Step 2: Expressing the Vector with Unit Vectors

In 3D space, any vector \( \langle x, y, z \rangle \) can be expressed using unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) as \( x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \). For the given vector, this becomes \( -a\mathbf{i} + \frac{1}{3}a\mathbf{j} + 4\mathbf{k} \).

Key Concepts

Unit Vectors

In the world of vectors, unit vectors play a vital role in defining directions. A unit vector is a vector with a magnitude of exactly 1. It is used to specify a direction without considering the length of the vector. In three-dimensional space, the standard unit vectors are represented as **i**, **j**, and **k**. Each of these vectors points in the direction of the respective axis:

• **i** points in the direction of the x-axis and is represented as \((1, 0, 0)\)
• **j** points in the direction of the y-axis and is represented as \((0, 1, 0)\)
• **k** points in the direction of the z-axis and is represented as \((0, 0, 1)\)

These unit vectors can be further used to construct any vector in the 3D space by scaling and vector addition. They provide a standardized way to express vectors, simplifying many vector operations.

Vector Components

Vectors in 3D can be expressed in terms of their components, which are projections of the vector along the x, y, and z axes. A vector like \(\langle x, y, z \rangle\) can be broken down into:

• **x**: The component along the x-axis.
• **y**: The component along the y-axis.
• **z**: The component along the z-axis.

This breakdown is crucial as it allows us to express the vector in a form that is easy to interpret and use in calculations: \(x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\). In this representation, you simply multiply each component by the corresponding unit vector. This way, each term represents a shift along one axis, and when combined, these shifts represent the original vector in the 3D space. It's like using a recipe that has ingredients (components) with precise amounts, ensuring the vector is accurately constructed from its smallest parts.

3D Vectors

A 3D vector is a directed line segment in a three-dimensional space, characterized by three components that determine its direction and length. This space is most commonly visualized with the help of the Cartesian coordinate system, where any vector can be described using components along the x-axis, y-axis, and z-axis: \(\langle x, y, z \rangle\).
3D vectors allow us to model real-world phenomena more accurately, as most objects and forces in the physical world operate in three dimensions rather than in two. By using the components \(x\), \(y\), and \(z\), we can effectively work with vectors in problem-solving, simulations, and other calculations in physics and engineering.
The sum of scaled unit vectors, \(x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\), forms the building block for representing any 3D vector. This makes it easier to perform vector operations, understand spatial orientation, and calculate vector properties like magnitude, dot products, and cross products.