Problem 50 - Precalculus

Question

Express the vector in terms of unit vectors i and \(j\) $$\langle 0,2\rangle$$

Step-by-Step Solution

Answer

The vector \(\langle 0, 2 \rangle\) is expressed as \(2j\).

Step 1: Understand Vector Components

The given vector is \(\langle 0, 2 \rangle\), which represents a two-dimensional vector with an x-component of 0 and a y-component of 2. We need to express this vector in terms of the unit vectors \(i\) and \(j\).

Step 2: Define Unit Vectors

The unit vector \(i\) is defined as \(\langle 1, 0 \rangle\) and represents a vector in the positive x-direction. The unit vector \(j\) is \(\langle 0, 1 \rangle\) and represents a vector in the positive y-direction.

Step 3: Rewrite the Vector

To express \(\langle 0, 2 \rangle\) in terms of \(i\) and \(j\), we multiply the x-component by \(i\) and the y-component by \(j\) and sum these results. This gives us: \(0 \cdot i + 2 \cdot j\).

Step 4: Simplify the Expression

Since the x-component is 0, the term \(0 \cdot i\) is 0. Therefore, the expression simplifies to \(2j\).

Key Concepts

Unit Vector

A unit vector is a special type of vector that has a magnitude of exactly one unit. It's usually employed to signify a specific direction in a coordinate system without affecting the magnitude of another vector it might interact with. There are basic unit vectors in two-dimensional space:

• Unit vector i: This vector, \( \langle 1, 0 \rangle\ \), points in the direction of the positive x-axis. It effectively captures movement along the x-direction without any movement in the y-direction.
• Unit vector j: This vector, \( \langle 0, 1 \rangle\ \), points in the direction of the positive y-axis. It represents movement along the y-direction without any movement in the x-direction.

Using these unit vectors, any vector in a two-dimensional space can be represented in terms of its direction. This is particularly valuable for simplifying vector calculations and expressing vectors in a standard form.

Vector Components

Every vector in a two-dimensional space can be broken down into two perpendicular components: the horizontal (x-direction) and the vertical (y-direction). These components act like building blocks that define the direction and magnitude of the vector in the plane. In mathematical terms:

• The x-component represents the vector's effect along the horizontal axis and is aligned with the unit vector \(i\).
• The y-component shows the vector's impact along the vertical axis, aligning with the unit vector \(j\).

When you know the components of a vector, you gain clear insights into how a vector behaves in each dimension. For instance, the vector \(\langle 0, 2 \rangle\\) has no x-component, indicating no movement in the horizontal direction, whereas its y-component of 2 shows movement purely upwards along the y-axis.

Two-dimensional Vector

Two-dimensional vectors describe quantities that have both direction and magnitude within a plane. They are expressed with two numerical components and greatly benefit from the unit vector notation, simplifying complex vector operations. Here’s why understanding two-dimensional vectors is essential:

• They can effectively represent real-world quantities such as displacement, velocity, and force within a plane.
• They provide clarity in spatial representations, being flexible in applications from physics to computer graphics.

By expressing a vector in terms of unit vectors like \(i\) and \(j\), it becomes easier to manipulate and understand how it interacts with other vectors. For example, \(\langle 0, 2 \rangle\\) can be rewritten using unit vector notation as \(2j\\), clearly highlighting the vector's direction and magnitude along the y-axis without any x-axis component. Understanding this form makes vector calculations more manageable and intuitive.