Problem 48 - Precalculus

Question

Express the vector in terms of unit vectors i and \(j\) $$(-6,-2)$$

Step-by-Step Solution

Answer

The vector is \(-6\mathbf{i} - 2\mathbf{j}\).

Step 1: Understand the Vector Notation

Vectors in 2D can be represented in terms of unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). The vector \((-6, -2)\) can be expressed as a combination of these unit vectors.

Step 2: Break Down the Components

Identify the components of the vector: \(-6\) for the \(x\)-component and \(-2\) for the \(y\)-component. This means the vector in terms of unit vectors is \(-6\mathbf{i}\) plus \(-2\mathbf{j}\).

Step 3: Write the Vector in Unit Vector Notation

Combine the values you've identified: the vector \((-6, -2)\) can be expressed as \(-6\mathbf{i} - 2\mathbf{j}\).

Key Concepts

Vector Notation

Vector notation is a useful way to describe both the direction and magnitude of physical quantities in mathematics and physics. In a two-dimensional space, you can represent vectors using ordered pairs, such as \((-6, -2)\). These pairs consist of two components:
- The first component corresponds to the direction and magnitude along the x-axis.- The second component corresponds to the direction and magnitude along the y-axis.
By using these components, we can effectively understand and perform operations with vectors. It's essential to grasp this notation, as it simplifies the communication and calculation of vectors. Vector notation facilitates expressing vectors in different forms, such as using unit vectors.

i and j Components

In vector notation, especially in 2D, unit vectors are represented by \(\mathbf{i}\) and \(\mathbf{j}\). These unit vectors have specific roles:
- \(\mathbf{i}\) denotes a unit vector along the x-axis. This means it has a magnitude of 1 in the x-direction.- \(\mathbf{j}\) denotes a unit vector along the y-axis, also having a magnitude of 1 in the y-direction.
When you express a vector using \(\mathbf{i}\) and \(\mathbf{j}\), you essentially break down the vector into its x and y components. For the vector \((-6, -2)\), it converts into \(-6\mathbf{i} - 2\mathbf{j}\). Here:
- \(-6\mathbf{i}\) indicates that the vector extends 6 units in the negative x-direction.- \(-2\mathbf{j}\) indicates that it extends 2 units in the negative y-direction.
This representation helps you to visualize vectors as sums of these perpendicular components.

2D Vectors

2D vectors are a fundamental concept in mathematics, influenced by their two-dimensional nature. A 2D vector describes a quantity that has both direction and magnitude in a plane. This distinction is critical in fields like physics, engineering, and graphics.
A typical 2D vector, like \((-6, -2)\), consists of two parts:

• An x-component, representing horizontal change.
• A y-component, representing vertical change.

These components let you understand how far and in which direction the vector moves from the origin (0,0).By dealing with 2D vectors, you can handle all vector-related tasks that occur on a flat surface, which is everything from calculating forces to simulating movements. Working with 2D vectors often involves converting between different forms, such as from geometric points to unit vector notation, enhancing flexibility and problem-solving.