91Ó°ÊÓ

In physics and mathematics, representing a vector in unit-vector notation is essential as it breaks down the vector into its fundamental components along the standard axes. This method uses unit vectors to signify direction.
- The unit vectors are: \( \hat{i} \) for the x-axis, \( \hat{j} \) for the y-axis, and \( \hat{k} \) for the z-axis.
With the given point \( (-5.0\, \mathrm{m}, 8.0\, \mathrm{m}, 0\, \mathrm{m}) \), we can express its position vector \( \mathbf{r} \) as follows in unit-vector notation:
\[ \mathbf{r} = -5.0 \hat{i} + 8.0 \hat{j} + 0 \hat{k} \]
This expression clearly shows how far and in which direction each component of the point is from the origin. Each term indicates displacement along the respective axes.

The magnitude of a vector tells us how long the vector is, quantifying its total length regardless of direction. Calculating this involves using the Pythagorean theorem in three dimensions.
For a given position vector \( \mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k} \), the formula to determine its magnitude is:
\[ |\mathbf{r}| = \sqrt{x^2 + y^2 + z^2} \]
For our position vector \( \mathbf{r} = -5.0 \hat{i} + 8.0 \hat{j} + 0 \hat{k} \), the magnitude can be calculated as:
\[ |\mathbf{r}| = \sqrt{(-5.0)^2 + (8.0)^2 + 0^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.43 \text{ m} \]
This result represents the straight-line distance from the origin to the point.

Understanding the angle a vector makes with an axis helps us interpret its orientation in space. Especially important is its angle relative to the x-axis which typically acts as a reference direction.
To determine this angle, we use trigonometry, specifically the arctangent function. For the vector \( \mathbf{r} \) given by its components \( x \) and \( y \), the angle \( \theta \) is calculated as:
\[ \theta = \tan^{-1}\left( \frac{y}{x} \right) \]
Applying this to our coordinates \( x = -5.0 \) and \( y = 8.0 \), we find:
\[ \theta \approx \tan^{-1}(\frac{8.0}{-5.0}) \approx -58.0^\circ \]
In vector terminology, a negative angle suggests passage into a different quadrant, hence recalculating to remain consistent with quadrant rules gives:
\[ 180.0^\circ - 58.0^\circ = 122.0^\circ \]
This means the vector is pointing 122 degrees counterclockwise from the positive x-axis.

The displacement vector represents the change in position of a point between two states. It's crucial because it not only indicates movement but also the direction of movement.
Starting from initial coordinates \( (-5.0 \mathrm{~m}, 8.0 \mathrm{~m}, 0 \mathrm{~m}) \) and moving to \( (3.00 \mathrm{~m}, 0 \mathrm{~m}, 0 \mathrm{~m}) \), the displacement vector \( \mathbf{d} \) is calculated as the difference between final and initial positions:
\[ \mathbf{d} = 3.00 \hat{i} - (-5.0 \hat{i}) + (0 \hat{j} - 8.0 \hat{j}) + (0 \hat{k} - 0 \hat{k}) \]
Which simplifies to:
\[ \mathbf{d} = (3.00 + 5.00) \hat{i} + (0 - 8.0) \hat{j} = 8.00 \hat{i} - 8.00 \hat{j} \]
The negative sign indicates the movement opposite to the initial direction along the y-axis. This vector not only tells us the shift in position but also the new direction at each point.