Problem 27 - Algebra and Trigonometry

Question

\(27-30\) Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\) . $$ \mathbf{u}=\langle 1,4\rangle $$

Step-by-Step Solution

Answer

The vector is \( \mathbf{u} = 1\mathbf{i} + 4\mathbf{j} \).

Step 1: Identify the Vector Components

The vector \( \mathbf{u} = \langle 1, 4 \rangle \) has two components. The first component is 1, and the second component is 4.

Step 2: Express using Unit Vectors

To write the vector \( \mathbf{u} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \), we use the formula \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \) where \( a \) is the first component and \( b \) is the second component.

Step 3: Substitute the Components into Unit Vectors

Substitute \( a = 1 \) and \( b = 4 \) into the formula: \( \mathbf{u} = 1\mathbf{i} + 4\mathbf{j} \).

Key Concepts

Vector Components

When we talk about vectors in mathematics, they are often described in terms of their components. Components are essentially the building blocks of a vector. For a two-dimensional vector, like the one given in the problem: \( \mathbf{u} = \langle 1, 4 \rangle \), these components represent how far the vector stretches in the horizontal and vertical directions of a Cartesian coordinate system.

• The first number inside the angle brackets, \(1\), is the horizontal component. This tells us how far and in which direction the vector travels left or right.
• The second number, \(4\), is the vertical component. This specifies how far and in which direction the vector moves up or down.

Together, these components comprehensively define the direction and magnitude of the vector.

Unit Vectors

Unit vectors are the basic building blocks of vector notation. They help describe vectors in terms of standard directions. Think of them as the compass directions of the mathematical world. In the context of our problem, the unit vectors are \( \mathbf{i} \) and \( \mathbf{j} \):

• \( \mathbf{i} \) represents a unit vector pointing in the horizontal direction, typically associated with the x-axis.
• \( \mathbf{j} \) is a unit vector pointing in the vertical direction, akin to the y-axis.

By expressing any vector in terms of these unit vectors, you can easily understand and communicate the vector's direction. For vector \( \mathbf{u} = \langle 1, 4 \rangle \), using unit vectors translates to \( \mathbf{u} = 1\mathbf{i} + 4\mathbf{j} \). This provides a systematic way to describe and manipulate vectors.

Two-Dimensional Vectors

Two-dimensional vectors are a special type of vector that exist in the plane defined by the x-axis and y-axis. They are handy for visualizing situations where directions and magnitudes in a plane are crucial, such as in wind speeds or forces acting on a point.

• They consist of two components: one horizontal and one vertical, like our vector \( \mathbf{u} = \langle 1, 4 \rangle \).
• In this simple 2D space, vectors can be effectively represented by using unit vectors to break down their components, simplifying comprehension and calculations.
• Graphically, they can be plotted on a plane and appear as arrows pointing from the origin to a specific point.

Understanding two-dimensional vectors is foundational for moving onto more complex multi-dimensional vectors and operations, making them a critical concept in vector mathematics.