Problem 19 - Algebra and Trigonometry

Question

Express the given vector in terms of the unit vectors i, j, and k. $$ \langle 12,0,2\rangle $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Components of the Vector

The vector given is \( \langle 12, 0, 2 \rangle \). This is a 3-dimensional vector with components along the x, y, and z axes.

Step 2: Assign Components to Unit Vectors

The components of the vector \( \langle 12, 0, 2 \rangle \) correspond to the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). Specifically, the first component (12) is along the \( \mathbf{i} \) direction, the second component (0) is along the \( \mathbf{j} \) direction, and the third component (2) is along the \( \mathbf{k} \) direction.

Step 3: Write the Vector in Terms of Unit Vectors

Using the unit vectors and their corresponding components, the vector can be expressed as:\[ 12\mathbf{i} + 0\mathbf{j} + 2\mathbf{k} \]Since \( 0\mathbf{j} \) contributes nothing, it can be omitted. Thus, the vector simplifies to:\[ 12\mathbf{i} + 2\mathbf{k} \]

Key Concepts

Unit Vectors

Unit vectors are fundamental in vector notation. They are vectors with a length of one and are used to represent direction along various axes in a coordinate system. In three-dimensional space, the unit vectors are denoted by \( \mathbf{i}, \mathbf{j}, \text{and} \mathbf{k} \). These represent the principal directions along the x, y, and z axes, respectively.

• \( \mathbf{i} \) points in the positive x-direction.
• \( \mathbf{j} \) points in the positive y-direction.
• \( \mathbf{k} \) points in the positive z-direction.

By using unit vectors, any vector can be expressed in terms of its components along these axes. This makes it easy to break down vectors into their directional parts, simplifying calculations and allowing for clear visualization of the vector's orientation.

Vector Components

Understanding vector components is crucial when working with vectors. A vector's components are essentially projections along the coordinate axes. Each component describes how much the vector stretches along a specific axis. For example, a vector \( \langle a, b, c \rangle \) has:

• An 'a' component along the x-axis (paired with \( \mathbf{i} \)).
• A 'b' component along the y-axis (paired with \( \mathbf{j} \)).
• A 'c' component along the z-axis (paired with \( \mathbf{k} \)).

By breaking a vector into components, you can better analyze its direction and magnitude in each dimension. This separation of components allows for simpler calculations, such as vector addition and multiplication, and is typical when converting from formal notation \( \langle a, b, c \rangle \) to expressions using unit vectors like \( a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \).

Three-Dimensional Vectors

Three-dimensional vectors are vectors that exist within three-dimensional space. These vectors have three components, corresponding to the three axes of the space: x, y, and z. A 3D vector can be expressed using notation \( \langle x, y, z \rangle \), which helps in determining the vector’s position and direction in space.
When working with 3D vectors:

• The first number usually represents the vector's extent along the x-axis.
• The second number represents the extent along the y-axis.
• The third number represents the extent along the z-axis.

Expressing these vectors using unit vectors like \( x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) provides clarity and precision. It’s a common method of visualization in physical sciences and engineering, where real-world objects are modeled in three dimensions. A 3D vector is powerful in conveying both magnitude and direction in a comprehensive form. By decomposing it into unit vectors, we can handle larger and more complex physical problems with ease.