91Ó°ÊÓ

A position vector is a fundamental concept in physics and mathematics. It defines the position of a particle or a point in a coordinate system.
In our example, the position vector \(\vec{\mathbf{r}}\) describes where a particle is located in relation to a chosen origin point. It usually has both magnitude and direction.
In terms of coordinates, if you know the x and y positions, you can write the position vector like this:

• \(\vec{\mathbf{r}} = x \mathbf{i} + y \mathbf{j}\)

Here, \(x\) and \(y\) represent the coordinates in a 2D space, while \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors along the x and y axes respectively.
For our exercise, the position vector changes with time \(t\), indicating that the particle moves over time.

The magnitude of a vector is a measure of its length or size, regardless of its direction. It's like asking, "How far does the vector reach from its starting point?"
Calculated using the Pythagorean theorem, the magnitude is always positive or zero.
To find the magnitude of a vector like \(\Delta \vec{\mathbf{r}} = 175 \mathbf{i} - 375 \mathbf{j}\), you use:

• \(|\Delta \vec{\mathbf{r}}| = \sqrt{(175)^2 + (-375)^2}\)

This formula squares each component, adds them together, and takes the square root of the result.
For the vector in our exercise:

• The magnitude is approximately \(413.61 \text{ m}\).

This shows how the vector stretches spatially between its start and endpoint.

Understanding the direction of a vector helps indicate where a vector is pointing in space.
It is typically defined as an angle relative to the positive x-axis and can be described using trigonometric functions.
For example, to calculate the direction of a vector \(\Delta \vec{\mathbf{r}} = 175 \mathbf{i} - 375 \mathbf{j}\), you find the angle \(\theta\):

• Use \(\tan \theta = \frac{-375}{175}\)
• Find \(\theta\) using the inverse tangent function: \(\theta = \arctan\left(\frac{-375}{175}\right)\)

The angle in our exercise was calculated to be approximately \(-64.8^\circ\).
This negative value means the vector points downward below the x-axis.
The direction complements the magnitude by telling not just how far, but where the displacement happens.