91Ó°ÊÓ

The magnitude of a vector gives us an idea of how long the vector is. Think of it as the 'length' from the start point to the endpoint. To find the magnitude, we'll use a special formula that involves the vector's components.

• Each vector has two parts: the x-component and the y-component.
• In our exercise, the position vector for the electron is \(\vec{r} = (5.0 \ \mathrm{m}) \hat{\mathrm{i}} - (3.0 \ \mathrm{m}) \hat{\mathrm{j}}\).

The magnitude formula is: \[ \| \vec{r} \| = \sqrt{(r_i)^2 + (r_j)^2} \]
Here, \( r_i \) and \( r_j \) are the x and y components, respectively. Just plug them into the formula and do the math! For our vector: \[ \| \vec{r} \| = \sqrt{(5.0)^2 + (-3.0)^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83 \mathrm{m} \]
So, the magnitude of our position vector is approximately 5.83 meters.

A coordinate system helps us visualize where things are in space. It mainly consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Where these two lines meet is called the origin (point \( (0,0) \)). Let's break this down:

• The x-axis shows how far left or right something is.
• The y-axis shows how far up or down something is.
• By combining these, we can pinpoint any location in our space.

To better understand, imagine a graph paper. The horizontal and vertical lines on the graph represent the coordinate system. Every point on the paper has an x-coordinate and a y-coordinate. These coordinates tell us exactly where the point is. In our exercise, we use this system to plot the vector \(\vec{r} = (5.0 \ \mathrm{m}) \hat{\mathrm{i}} - (3.0 \ \mathrm{m}) \hat{\mathrm{j}}\).
To sketch this:

• Start at the origin \( (0,0) \).
• Move 5.0 meters to the right (positive x-direction).
• Then move 3.0 meters down (negative y-direction).
• Draw an arrow from the origin to the final point \( (5.0, -3.0) \).

This is your position vector!

Vector components break a vector into its basic parts along the coordinate axes. This makes calculations simpler and more understandable.
A vector in two dimensions can be split into:

• The x-component: Shows how far the vector goes along the x-axis.
• The y-component: Shows how far the vector goes along the y-axis.

For our vector, \(\vec{r} = (5.0 \ \mathrm{m}) \hat{\mathrm{i}} - (3.0 \ \mathrm{m}) \hat{\mathrm{j}}\), the two components are:

• X-component: \( 5.0 \mathrm{m} \)
• Y-component: \( -3.0 \mathrm{m} \)

Each component helps us understand how the vector stretches along the x and y directions. By knowing these, we can reconstruct the vector using these building blocks.
For instance, if we metaphorically 'move' 5.0 meters to the right and then 3.0 meters down, we get the same effect as moving along the vector \(\vec{r} \). Easy, right?

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides in a right triangle. It states: \[ c^2 = a^2 + b^2 \]
Where:

• \( c \) is the hypotenuse (the side opposite the right angle).
• \( a \) and \( b \) are the other two sides.

This theorem is crucial for finding the magnitude of a vector. Imagine a right triangle formed by the vector's components along the x and y axes. The vector itself represents the hypotenuse! In our exercise, we have: \[ a = 5.0 \mathrm{m}, b = -3.0 \mathrm{m} \] Using the Pythagorean theorem, the magnitude of the vector (hypotenuse \( c \)) is: \[ c = \sqrt{a^2 + b^2} = \sqrt{(5.0)^2 + (-3.0)^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83 \mathrm{m} \]
This theorem helps simplify obtaining the vector's magnitude and offers a straightforward method to understand vector calculations. Remember, any time you see a right triangle, think of the Pythagorean theorem!