Problem 24 - Precalculus Mathematics for Calculus

Question

Express the given vector in terms of the unit vectors i, \(\mathbf{j}\). and \(\mathbf{k}\). $$\mathbf{u}=\langle 3,1,0\rangle, \quad \mathbf{v}=\langle 3,0,-5\rangle$$

Step-by-Step Solution

Answer

\( \mathbf{u} = 3\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} \), \( \mathbf{v} = 3\mathbf{i} - 5\mathbf{k} \).

Step 1: Understand the Unit Vectors

The unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) are the standard basis vectors in three-dimensional space. Specifically, \( \mathbf{i} = \langle 1, 0, 0 \rangle \), \( \mathbf{j} = \langle 0, 1, 0 \rangle \), and \( \mathbf{k} = \langle 0, 0, 1 \rangle \). These vectors represent directions along the x-axis, y-axis, and z-axis, respectively.

Step 2: Identify Components of Vector \( \mathbf{u} \)

The vector \( \mathbf{u} = \langle 3, 1, 0 \rangle \) can be broken down into its components: the x-component is 3, the y-component is 1, and the z-component is 0.

Step 3: Express \( \mathbf{u} \) in Terms of Unit Vectors

Using the components identified, express \( \mathbf{u} \) in terms of unit vectors: \( \mathbf{u} = 3\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} \).

Step 4: Identify Components of Vector \( \mathbf{v} \)

The vector \( \mathbf{v} = \langle 3, 0, -5 \rangle \) can be broken down into its components: the x-component is 3, the y-component is 0, and the z-component is -5.

Step 5: Express \( \mathbf{v} \) in Terms of Unit Vectors

Using the components identified, express \( \mathbf{v} \) in terms of unit vectors: \( \mathbf{v} = 3\mathbf{i} + 0\mathbf{j} - 5\mathbf{k} \).

Key Concepts

Unit Vectors

Unit vectors are fundamental in understanding and working with vectors in any space. They provide a standard way to reference directions without concerning magnitude. In three-dimensional space, the commonly used unit vectors are \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).

These unit vectors are defined as follows:

• \( \mathbf{i} = \langle 1, 0, 0 \rangle \) - corresponds to the x-axis direction.
• \( \mathbf{j} = \langle 0, 1, 0 \rangle \) - corresponds to the y-axis direction.
• \( \mathbf{k} = \langle 0, 0, 1 \rangle \) - corresponds to the z-axis direction.

These vectors have a magnitude of 1, which is why they are called unit vectors. When expressing a vector in terms of these unit vectors, we essentially decompose the vector into its components along each axis.

Vector Components

Vector components are essentially the building blocks that define a vector in any space. For a three-dimensional vector like \( \mathbf{u} = \langle 3, 1, 0 \rangle \), the components represent how far the vector stretches along each of the respective axes: x, y, and z.

• The x-component of \( \mathbf{u} \) is 3, indicating it moves three units in the direction of the x-axis.
• The y-component is 1, showing a single unit movement along the y-axis.
• The z-component is 0, suggesting that there is no movement in the z-direction.

By expressing a vector such as \( \mathbf{u} \) as \( 3\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} \), we unravel it into its components, making it far more manageable to visualize and use in calculations.

Basis Vectors

Basis vectors form the foundation for a vector space, providing a frame of reference where any vector can be expressed. In a three-dimensional space, the set of unit vectors \( \{\mathbf{i}, \mathbf{j}, \mathbf{k}\} \) can act as basis vectors.

When you express a vector like \( \mathbf{v} = \langle 3, 0, -5 \rangle \) in terms of these basis vectors:

• \( \mathbf{v} = 3\mathbf{i} \) means 3 units in the x-direction.
• \( + 0\mathbf{j} \) implies no movement in the y-direction.
• \( - 5\mathbf{k} \) depicts a movement of 5 units in the opposite z direction.

This representation allows vectors to be expressed in a consistent, standardized way. This is crucial for operations like vector addition, subtraction, and scalar multiplication, facilitating easy manipulation and calculation in vector algebra.