Final answer:
To find a vector from point A to B, calculate the differences in the x and y coordinates to obtain the components of the vector AB. Then, find the magnitude and direction using the Pythagorean theorem and arctan function, respectively.
Step-by-step explanation:
Finding a Vector in the Direction of a Line Segment
To find a vector in the direction of the line segment from point A to point B, we first define these points in Cartesian coordinates, say A(x1, y1) and B(x2, y2). The vector in the direction from A to B is obtained by subtracting the coordinates of A from those of B, which can be written as vector AB = B - A. Mathematically, this procedure involves calculating the differences in the x and y components separately:
- ABx = x2 - x1
- ABy = y2 - y1
This results in a new vector, AB, with components (ABx, ABy). This vector points from A to B and has both magnitude and direction. The magnitude of AB can be calculated using the Pythagorean theorem, √(ABx² + ABy²), and the direction can be determined using trigonometry, for instance, by calculating the angle θ with respect to the x-axis using the arctan function: θ = arctan(ABy/ABx). The graphical method involves plotting this vector on a graph and using tools like a ruler and protractor for a visual representation.