Draw the following vectors in standard position in ℝ^2 : (a)...

Draw the following vectors in standard position in $\mathbb{R}^{2}$ : (a) $a=\left[\begin{array}{l}3 \\ 0\end{array}\right]$ (b) $\mathbf{b}=\left[\begin{array}{l}2 \\ 3\end{array}\right]$ (c) $c=\left[\begin{array}{r}-2 \\ 3\end{array}\right]$ (d) $\mathbf{d}=\left[\begin{array}{r}3 \\ -2\end{array}\right]$

Draw the following vectors in standard position in $\mathbb{R}^{2}$ :
(a) $a=\left[\begin{array}{l}3 \\ 0\end{array}\right]$
(b) $\mathbf{b}=\left[\begin{array}{l}2 \\ 3\end{array}\right]$
(c) $c=\left[\begin{array}{r}-2 \\ 3\end{array}\right]$
(d) $\mathbf{d}=\left[\begin{array}{r}3 \\ -2\end{array}\right]$

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Key Concepts

Cartesian Coordinate System

The Cartesian coordinate system consists of a horizontal axis (x-axis) and a vertical axis (y-axis), which intersect at the origin. This system provides a framework for plotting vectors by using their x and y components to determine precise positions in the plane.

Component Form

Component form refers to expressing a vector by its individual components along the coordinate axes. This form, typically written as [x, y] in ?², clearly shows the contributions of each axis to the vector's overall direction and magnitude.

Vector

A vector is a mathematical object that has both magnitude and direction. In ?², vectors are often represented as ordered pairs (x, y), where the first component represents the horizontal displacement and the second represents the vertical displacement.

Standard Position

Placing a vector in standard position means positioning its tail at the origin (0, 0) of the coordinate system so that its head represents the terminal point. This standardization simplifies comparison between vectors and facilitates operations such as addition or scaling.

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