91Ó°ÊÓ

A position vector is a crucial concept in vector mathematics. It indicates the position of a point in space relative to an origin.
In the exercise, the position vector of the electron is represented as \( \vec{r} = (5.0 \, \mathrm{m}) \hat{\mathrm{i}} - (3.0 \, \mathrm{m}) \hat{\mathrm{j}} + (2.0 \, \mathrm{m}) \hat{\mathrm{k}} \).
This expression breaks down into components:

• \( (5.0 \, \mathrm{m}) \hat{\mathrm{i}} \) represents 5 meters in the x-direction.
• \( (-3.0 \, \mathrm{m}) \hat{\mathrm{j}} \) represents -3 meters in the y-direction.
• \( (2.0 \, \mathrm{m}) \hat{\mathrm{k}} \) represents 2 meters in the z-direction.

Each term describes how far and in which direction the position is from the origin. Understanding position vectors aids in visualizing where an object is located in a 3D space.

The magnitude of a vector is akin to measuring its length from start to end. For a position vector like \( \vec{r} = (5.0 \, \mathrm{m}) \hat{\mathrm{i}} - (3.0 \, \mathrm{m}) \hat{\mathrm{j}} + (2.0 \, \mathrm{m}) \hat{\mathrm{k}} \), finding its magnitude involves calculating the distance.To determine the magnitude \( |\vec{r}| \), utilize the formula:\[|\vec{r}| = \sqrt{x^2 + y^2 + z^2}\]Substituting the components of the position vector gives:\[|\vec{r}| = \sqrt{(5.0)^2 + (-3.0)^2 + (2.0)^2} = \sqrt{25 + 9 + 4} = \sqrt{38}\]Thus, the magnitude is \( \sqrt{38} \approx 6.16 \mathrm{\ m} \).
This value tells you how far the electron is from the origin in 3D space. Understanding the magnitude helps in comparing vectors and understanding spatial relationships.

In physics and mathematics, coordinate systems are frameworks that allow us to express the positions of points in space.
A common coordinate system is the Cartesian coordinate system, or rectangular coordinate system, often described by three perpendicular axes labeled x, y, and z.
This is known as a right-handed coordinate system because if you curl the fingers of your right hand from the x-axis towards the y-axis, your thumb points in the direction of the z-axis.In the problem, the position vector is located within this right-handed coordinate system. This means:

• The x-component (\(5.0 \mathrm{\ m}\)) lies along the horizontal direction.
• The y-component (\(-3.0 \mathrm{\ m}\)) is aligned vertically but in the opposite direction of the positive y-axis.
• The z-component (\(2.0 \mathrm{\ m}\)) extends in a direction perpendicular to both x and y axes.

Understanding coordinate systems allows for interpreting vectors in three-dimensional space and simplifies complex physical problems by classifying positions and directions with respect to the origin.