Problem 29 - Algebra and Trigonometry

Question

\(27-30\) Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\) . $$ \mathbf{u}=\langle 3,0\rangle $$

Step-by-Step Solution

Answer

The vector is \( \mathbf{u} = 3 \mathbf{i} \).

Step 1: Understand the vector components

The vector \( \mathbf{u} = \langle 3, 0 \rangle \) is given in component form, where 3 is the horizontal component (\( x \)-component) and 0 is the vertical component (\( y \)-component).

Step 2: Identify the unit vectors

In vector notation, \( \mathbf{i} \) represents the unit vector in the horizontal direction, and \( \mathbf{j} \) represents the unit vector in the vertical direction.

Step 3: Express the vector using unit vectors

To write the vector \( \mathbf{u} = \langle 3, 0 \rangle \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \), multiply the \( x \)-component by \( \mathbf{i} \) and the \( y \)-component by \( \mathbf{j} \): \[ \mathbf{u} = 3 \mathbf{i} + 0 \mathbf{j} \]

Step 4: Simplify the expression

Since multiplying by zero results in zero, the term \( 0 \mathbf{j} \) is equivalent to zero. Therefore, the vector can be simplified to: \[ \mathbf{u} = 3 \mathbf{i} \]

Key Concepts

Vector Components

Vector components are the building blocks of a vector. When a vector is expressed in component form, like \( \mathbf{u} = \langle 3, 0 \rangle \), it includes the vector's magnitude along the horizontal (\( x \)-axis) and vertical (\( y \)-axis) directions.
Each value in this pair represents how far and in which direction the vector stretches along each axis. For \( \mathbf{u} \), the number 3 is the horizontal component, meaning the vector stretches 3 units to the right along the \( x \)-axis, and 0 is the vertical component, implying it has no vertical stretch.
Understanding vector components helps in visually deconstructing the vector and offers clarity on how it behaves in a two-dimensional plane. Thus, breaking vectors into components simplifies many mathematical operations, as each component can be dealt with individually.

Unit Vectors

Unit vectors are fundamental in vector notation, providing direction within a vector space without influencing its magnitude. These vectors have a length of one unit. In a coordinate system, they serve as the basis for vector representation.
The two most common unit vectors are \( \mathbf{i} \) and \( \mathbf{j} \). The unit vector \( \mathbf{i} \) represents the horizontal direction along the \( x \)-axis, and \( \mathbf{j} \) points vertically along the \( y \)-axis.
When describing a vector using unit vectors, you indicate the vector's relative movement along each axis. For example, in the exercise, to express \( \mathbf{u} = \langle 3, 0 \rangle \) using unit vectors, the horizontal component (3) is multiplied by \( \mathbf{i} \) and the vertical component (0) by \( \mathbf{j} \).

• Horizontal representation: 3 times \( \mathbf{i} \)
• Vertical representation: 0 times \( \mathbf{j} \)

This helps create the vector's actual behavior in terms of direction and magnitude.

Vector Notation

Vector notation is a method to represent and simplify the expression of vectors. Vectors can be denoted in component form, such as \( \langle x,y \rangle \), or using unit vectors for clarity and conciseness.
In unit vector notation, any vector \( \mathbf{v} \) in two-dimensional space can be expressed as a linear combination of \( \mathbf{i} \) and \( \mathbf{j} \): \[ \mathbf{v} = a \mathbf{i} + b \mathbf{j} \]
Here, \( a \) and \( b \) are the coefficients corresponding to the vector's components along the \( x \)-axis and \( y \)-axis, respectively. This method allows you to see clearly how much influence each direction has within the vector's overall configuration.
For instance, in the exercise provided, the vector \( \mathbf{u} = \langle 3, 0 \rangle \) has been written as \( 3 \mathbf{i} + 0 \mathbf{j} \), simplifying to \( 3 \mathbf{i} \) because the term involving \( \mathbf{j} \) vanishes (due to multiplication by zero). This showcases not only the magnitude but also the direction in which the vector points.