Question

Vectors \(\vec{x}\) and \(\vec{y}\) are given. Sketch \(\vec{x}, \vec{y}, \vec{x}+\vec{y},\) and \(\vec{x}-\vec{y}\) on the same Cartesian axes. $$ \vec{x}=\left[\begin{array}{c} -1 \\ 1 \end{array}\right], \vec{y}=\left[\begin{array}{c} -2 \\ 2 \end{array}\right] $$

Short answer

Sketch the vectors \(\vec{x}, \vec{y}, \vec{x} + \vec{y},\) and \(\vec{x} - \vec{y}\) on the same axes based on calculated coordinates.

Step-by-step solution

Understand the Vectors

Identify the given vectors \(\vec{x}\) and \(\vec{y}\). Here, \(\vec{x} = \begin{bmatrix} -1 \ 1 \end{bmatrix}\) and \(\vec{y} = \begin{bmatrix} -2 \ 2 \end{bmatrix}\). Each vector is represented in two-dimensional space with an x-coordinate and y-coordinate.

Calculate \(\vec{x} + \vec{y}\)

Addition of vectors is done component-wise: \(\vec{x} + \vec{y} = \begin{bmatrix} -1 \ 1 \end{bmatrix} + \begin{bmatrix} -2 \ 2 \end{bmatrix} = \begin{bmatrix} -1 + (-2) \ 1 + 2 \end{bmatrix} = \begin{bmatrix} -3 \ 3 \end{bmatrix}\).

Calculate \(\vec{x} - \vec{y}\)

Subtraction of vectors is also done component-wise: \(\vec{x} - \vec{y} = \begin{bmatrix} -1 \ 1 \end{bmatrix} - \begin{bmatrix} -2 \ 2 \end{bmatrix} = \begin{bmatrix} -1 - (-2) \ 1 - 2 \end{bmatrix} = \begin{bmatrix} 1 \ -1 \end{bmatrix}\).

Graph the Vectors

To sketch the vectors on Cartesian axes: start the tail of each vector at the origin (0,0), and their heads at their respective coordinates. \(\vec{x}\) from \((0,0)\) to \((-1,1)\), \(\vec{y}\) from \((0,0)\) to \((-2,2)\), \(\vec{x} + \vec{y}\) from \((0,0)\) to \((-3,3)\), and \(\vec{x} - \vec{y}\) from \((0,0)\) to \((1,-1)\).

Analyze Your Sketch

In your sketch, observe how vector addition combines trends in both the runs (horizontal) and rises (vertical) of \(\vec{x}\) and \(\vec{y}\). Similarly, note how vector subtraction directs \(\vec{x}\) in a manner opposite to \(\vec{y}\). Ensure each vector's direction and length reflect calculations accurately.

Key concepts

Vector Addition

Vector addition is a core operation in vector algebra, and it's both fascinating and straightforward! The process happens component by component, meaning you add the corresponding coordinates of the vectors involved.

For instance, if you have vectors \(\vec{x} = \begin{bmatrix} -1 \ 1 \end{bmatrix}\) and \(\vec{y} = \begin{bmatrix} -2 \ 2 \end{bmatrix}\), their sum \(\vec{x} + \vec{y}\) is calculated by adding their respective components:

\(\vec{x} + \vec{y} = \begin{bmatrix} -1 + (-2) \ 1 + 2 \end{bmatrix} = \begin{bmatrix} -3 \ 3 \end{bmatrix}\).

This component-wise addition illustrates how the new vector \(\vec{x} + \vec{y}\) interacts in the Cartesian plane. The operation effectively results in a diagonal traversal across the plane, as seen by the individual changes in horizontal and vertical components from the origin.

Vector Subtraction

Vector subtraction, like vector addition, occurs on a component-by-component basis. Here, however, we subtract the components. If delving into the same vectors \(\vec{x} = \begin{bmatrix} -1 \ 1 \end{bmatrix}\) and \(\vec{y} = \begin{bmatrix} -2 \ 2 \end{bmatrix}\), the subtraction \(\vec{x} - \vec{y}\) follows this calculation:

\(\vec{x} - \vec{y} = \begin{bmatrix} -1 - (-2) \ 1 - 2 \end{bmatrix} = \begin{bmatrix} 1 \ -1 \end{bmatrix}\).

Essentially, you "reverse" the direction of \(\vec{y}\) and combine it with \(\vec{x}\). This gives a new direction and length character to the resultant vector that points from the head of \(\vec{y}\) towards the head of \(\vec{x}\) relative to the origin. Subtraction in vectors is akin to adding a vector pointing in the opposite direction.

Cartesian Plane

The Cartesian plane is where vectors are graphically represented, forming the bridge between algebraic computations and spatial visualization. It's a two-dimensional space defined by an x-axis (horizontal) and a y-axis (vertical).

In exercises involving vector algebra like our example, start graphing vectors from the origin (0,0). Each vector stands represented as directed line segments:

• \(\vec{x}\) extends from \((0,0)\) to \((-1,1)\).
• \(\vec{y}\) extends from \((0,0)\) to \((-2,2)\).
• \(\vec{x} + \vec{y}\) moves to \((-3,3)\).
• \(\vec{x} - \vec{y}\) stretches to \((1,-1)\).

This visual setup is crucial as it offers a tangible glimpse into the interactions and resultant directions vectors follow after operations.

Two-Dimensional Vectors

Two-dimensional vectors, simply known as 2D vectors, have the unique quality of expressing magnitude and direction with just two numbers - often referred to as components.

These components are traditionally labeled as the x-component and the y-component. For example, in \(\vec{x} = \begin{bmatrix} -1 \ 1 \end{bmatrix}\), \(-1\) is its x-component, while \(1\) is its y-component.

2D vectors are particularly useful when exploring concepts in physics or engineering across flat planes. They allow for mapping everything from forces acting on an object to the trajectory of projectiles. Understanding their basic operations—like addition and subtraction—lays the foundation for mastering more complex topics in vector analysis and geometry.