Problem 20 - Algebra and Trigonometry

Question

Express the given vector in terms of the unit vectors i, j, and k. $$ \langle 0,-3,5\rangle $$

Step-by-Step Solution

Answer

-3j + 5k

Step 1: Identify Components

The given vector is \( \langle 0, -3, 5 \rangle \). This means it has three components: the x-component is 0, the y-component is -3, and the z-component is 5.

Step 2: Express in Terms of Unit Vectors

A vector \( \langle a, b, c \rangle \) can be expressed in terms of the unit vectors \( \mathbf{i, j, k} \) as \( a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \). For our vector, this becomes \( 0\mathbf{i} - 3\mathbf{j} + 5\mathbf{k} \).

Step 3: Simplify Expression

Since multiplying a vector by zero results in a zero vector, \( 0\mathbf{i} \) can be omitted. Therefore, the vector expression simplifies to \( -3\mathbf{j} + 5\mathbf{k} \).

Key Concepts

Unit Vectors

Unit vectors are the building blocks of vector notation. They are vectors of length one that point along the axes of a coordinate system. In three-dimensional space, we usually denote unit vectors as \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).

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

These vectors have a magnitude of 1, and they are used to express any vector in three-dimensional space as a combination of these unit vectors.
This approach makes it easier to handle and visualize vectors, as each component of a vector aligns with one of these principal directions.

Components of a Vector

Each vector in a 3D space can be broken down into its components along the x, y, and z axes. To understand this better, consider the vector \( \langle 0, -3, 5 \rangle \) from our exercise.

• x-component: This is 0, indicating that along the x-axis, there's no movement.
• y-component: This is -3, suggesting the vector moves 3 units in the negative y direction.
• z-component: This is 5, showing that it extends 5 units in the positive z direction.

The components of a vector represent how much of the vector is aligned along each respective axis. They give a clear picture of the vector's direction and magnitude in each dimension.
When expressed in terms of unit vectors, each component gets multiplied by its respective unit vector. This converts the components into a more practical and readable form, using \( a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \), where \( a, b, \) and \( c \) are the components of the vector.

Simplifying Vectors

Simplifying vectors is often necessary to make expressions cleaner and more readable.
Sometimes, the original expression of a vector may contain unnecessary terms, like multiplying a unit vector by zero. Consider the vector \( 0\mathbf{i} - 3\mathbf{j} + 5\mathbf{k} \). The term \( 0\mathbf{i} \) contributes nothing since multiplying by zero results in zero.To simplify:

• Identify terms: Look for any terms multiplied by zero, as they can be eliminated.
• Eliminate zero terms: In this case, remove \( 0\mathbf{i} \) because it adds no value.
• Rewrite the vector: After removing unnecessary terms, you get \( -3\mathbf{j} + 5\mathbf{k} \), which is a much cleaner expression of the vector.

Simplifying not only clarifies the expression but also sometimes aids in calculations. It focuses on the components that actually contribute to the vector's direction and length.