# How Do You Calculate the Position Vector of Point P in a Given Ratio?

• Random-Hero-
In summary, to find the position vector of point P, we use the ratio 3:5 to determine its location between points A and B. This means that P is located 3/5 of the way from A to B. To find the coordinates of P, we take the coordinates of A and add 3/5 of the difference between A and B. This results in the coordinates (2,3,1) for point P. This method is accurate and can be used to find any point that divides a line segment in a given ratio.
Random-Hero-

## Homework Statement

Use vectors to find the position vector of point P if P divides AB in the ratio 3:2 given A (-1,6,4) and B (4,1,-1)

## The Attempt at a Solution

Well I assume it's sort of like finding the midpoint, except I would do something like (-1+4)/1.5 etc.

Am I correct? Do I just divide the sum of the numbers by 3/2 hence the ratio? or am I going about this the wrong way?

Point P is 3/5 the way along from A to B. When you divide something in the ratio 3:2, one part will be 3/5 of the total and the other will be 2/5 of the total.

It shouldn't be too hard to find that point.
Assuming for the moment that the coordinates of P are (x, y, z), the position vector of point P will be the vector (x - (-1), y - 6, z - 4). This vector has the same direction as AP and the same magnitude.

Is this a descent answer?

If AP:PB=3:2 then AP:AB=3:(3+2)=3:5
P = A + (B-A)*3/5
P = (-1,6,4)+(5,-5,-5)*3/5
P = (-1,6,4)+(3,-3,-3)
P = (2,3,1)

Am I correct? Can anybody clarify?

## 1. What is a position vector with ratio?

A position vector with ratio is a mathematical concept used to represent the position of a point in a coordinate system. It consists of a magnitude (or length) and a direction, expressed as a ratio of two numbers. This ratio is typically written in the form (x:y) where x represents the distance along the horizontal axis and y represents the distance along the vertical axis.

## 2. How is a position vector with ratio different from a regular position vector?

A regular position vector is expressed in terms of its components (x and y coordinates) while a position vector with ratio is expressed in terms of a ratio of two numbers. This allows for a more general representation of position, as it is not limited to a specific coordinate system. Additionally, the magnitude of a position vector with ratio can be easily changed by multiplying the ratio by a constant, while the magnitude of a regular position vector is fixed.

## 3. What is the purpose of using a position vector with ratio?

The purpose of using a position vector with ratio is to have a more flexible and general representation of position in mathematics. It can be used in any coordinate system, making it easier to perform calculations and transformations. It also allows for a more intuitive understanding of the position of a point, as the ratio provides a visual representation of the direction and distance from the origin.

## 4. How is a position vector with ratio calculated?

A position vector with ratio is calculated by finding the ratio of the distance from the point to the origin along each axis. For example, if a point has coordinates (3,4) in a Cartesian coordinate system, the position vector with ratio would be (3:4). This ratio can also be simplified, if needed, by dividing both numbers by their greatest common factor.

## 5. Can a position vector with ratio be used in three-dimensional space?

Yes, a position vector with ratio can be used in three-dimensional space. In this case, the ratio would be represented in the form (x:y:z), where x represents the distance along the x-axis, y represents the distance along the y-axis, and z represents the distance along the z-axis. This allows for a more general representation of position in three-dimensional space.

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