Question

Sketch the given vector field over the rectangle with opposite corners (-2,-2) and \((2,2),\) sketching one vector for every point with integer coordinates (i.e., at \((0,0),(1,2),\) etc. \()\) \(\vec{F}=\langle 1,-1\rangle\)

Short answer

The vectors in the field all point diagonally to the right and downwards.

Step-by-step solution

Understand the Vector Field

The given vector field is \( \vec{F} = \langle 1, -1 \rangle \). This means that at every point in the plane, the vector has a horizontal component of 1 and a vertical component of -1.

Identify Integer Coordinate Points

Identify all the points with integer coordinates within the rectangle with opposite corners at \((-2, -2)\) and \((2, 2)\). These points include: \((-2, -2), (-2, -1), (-2, 0), ..., (2, 2)\). Altogether, there are 25 such points.

Sketch the Vector at a Point

At each integer coordinate, sketch the vector \( \langle 1, -1 \rangle \). This vector moves one unit to the right and one unit down from the given point. Start sketching at the bottom-left of the rectangle and proceed to the top-right.

Complete the Sketch

Repeat the previous step for every integer coordinate point identified in Step 2. Ensure that each vector points in the same direction, showing the uniform nature of the vector field across the rectangle.

Key concepts

Integer Coordinate Points

Integer coordinate points are locations on a plane where both the x and y values are whole numbers. In the context of vector fields, they are crucial for plotting because they offer clear reference points for placing vectors.
To determine these integer coordinate points within a specific region, you examine all combinations of whole numbers within the given range. For example, in the problem of sketching a vector field over a rectangle with corners at (-2, -2) and (2, 2), integer points would include every possible pair of integers from -2 to 2. This results in 25 points as each axis provides 5 integer values.
Finding these points involves listing all integer pairs from corner to corner, such as:

• (-2, -2) to (-2, 2)
• (-1, -2) to (-1, 2)
• ... and so on until (2, 2)

You can imagine these points forming a grid, allowing you to systematically place each vector according to the instructions of the vector field.

Vector Components

Vector components are the individual directions a vector moves along the axes of a coordinate system. For example, a vector \(\vec{F} = \langle 1, -1 \rangle\) has two components. This means it moves 1 unit to the right (the x-component) and 1 unit down (the y-component).
Understanding vector components is essential as they dictate the direction and magnitude of the vector. Magnitude defines how much movement occurs and, since we're dealing with uniform fields, affects every point similarly.
To break down the process:

• The x-component (here 1) incrementally shifts each point horizontally.
• The y-component (here -1) decreases the vertical position.

This consistent movement shows that the vector field is uniform. Regardless of which integer point you start at, the vector \(\langle 1, -1 \rangle\) always points in the same direction, aligning each initial point with its vector-determined position.

Vector Sketching

Vector sketching involves drawing vectors at specific points to represent a vector field visually. In this exercise, it means placing a small arrow or line segment starting from each integer coordinate point.
When sketching from the point \((-2, -2)\) or any other point within the specified region, draw the vector based on its components. For \(\vec{F} = \langle 1, -1 \rangle\), begin where x and y intersect and direct the arrow 1 unit right and 1 unit down. Repeat this for every integer coordinate.
Here are some tips for effective sketching:

• Start from each integer coordinate listed, drawing vectors consistently.
• Keep vector size proportional to component values to maintain clarity.
• Ensure all vectors maintain their direction to reflect field uniformity.

The goal is to form an illustrative field pattern with these vectors, creating a network of aligned arrows indicating uniform movement, thereby visualizing the vector field accurately.