Question

Find the following vectors. The position vector for your final location if you start at the origin and walk along (4,-6) followed by \langle 5,9\rangle

Short answer

Answer: The final position vector is <9, 3>.

Step-by-step solution

Understanding the vectors

First, let's understand and represent the vectors properly. Vector 1: (4, -6) Vector 2: <5, 9>

Visualize walking from origin point by following the given vectors

To find the final position vector, imagine that you are walking from the origin point (0,0). First, you walk along Vector 1 that takes you to the point (4, -6). Next, you walk along Vector 2 that takes you to the new point from your current position.

Calculate the final location coordinates

To calculate the final location coordinates, you can sum up the corresponding coordinates of the two vectors. Final x-coordinate: x1 + x2 = 4 + 5 = 9 Final y-coordinate: y1 + y2 = (-6) + 9 = 3

Write down the position vector for the final location

Now that we have calculated the final location coordinates (9, 3), we can represent it as a position vector for your final location. Final position vector: <9,3>

Key concepts

Position Vector

Let's dive into the fascinating concept of a position vector! A position vector is a mathematical term used to describe the location of a point in space relative to a specific point known as the origin. Imagine a point as a destination on a map. The position vector shows how to reach that destination from the origin. It provides a directional arrow in a coordinate system and has both magnitude (length) and direction.

In the exercise, when you start at the origin and move according to given vectors, you trace out a path. Each vector specifies a direction and a distance. By summing these vectors, you compute the final position vector, which points directly from the origin to your final location.

Remember:

• A position vector is often represented by brackets, such as \(\), indicating the x-coordinate and y-coordinate where the point lies.
• It begins at the origin and ends at the given point.

Understanding position vectors helps simplify calculations and visualize movement in space.

Coordinate System

A coordinate system is a framework used to define and locate points in space. Think of it like a grid superimposed on a map that helps you pinpoint exactly where you are. It allows for consistency and precision in describing the position of points.

The most common type of coordinate system in these exercises is the Cartesian coordinate system, which consists of perpendicular axes: the x-axis (horizontal) and y-axis (vertical). The intersection of these axes, at (0, 0), is called the origin.

Key features of a coordinate system include:

• The ability to graphically represent vectors and points.
• Each point in the plane is given by an ordered pair (x, y).
• Facilitating operations like vector addition by graphically showing paths and endpoints.

Understanding coordinate systems is crucial for effectively working with vectors, as it lends structure to the calculations and visual reasoning.

Origin

The origin is a fundamental point in any coordinate system. It is the starting point, marked as (0, 0) in a 2-dimensional space, where the x-axis and y-axis intersect. Imagine it as the point of reference from which all other points are measured.

In many problems involving vectors, like the one from the exercise, you begin your journey from the origin. This point is crucial because it's the base reference for all position vectors. When the problem asks you to start at the origin, it means this is your initial location before any movements described by vectors.

Why the origin is important:

• It acts as a fixed starting point for measuring distances and directions.
• It simplifies calculations since all vectors are positioned relative to this point.
• Provides clarity and consistency, unifying various mathematical concepts around it.

Mastering the concept of the origin helps you grasp more complex vector operations and spatial relationships, making it a pillar of understanding in mathematics.