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The journey of an object, such as a positron, often begins with an initial position vector. Think of this as the starting point in a three-dimensional space. This vector \( \vec{r}_0 \) serves as a precise location from which any movement or displacement begins.
When we talk about vectors in physics, we refer to directions and magnitudes. The initial position vector gives us the direction and location where the object was present before it began moving. Visualizing the vector as an arrow can help. The tail of the arrow is at the origin, and its head points towards the object's starting position.

• The vector is expressed in terms of its components along the x, y, and z-axes. For example, \( \vec{r}_0 = -2.0 \hat{\mathrm{i}} + 6.0 \hat{\mathrm{j}} - 10.0 \hat{\mathrm{k}} \) states the initial positions along three perpendicular axes.
• Each component represents how far along each axis the object is from the origin.

From the exercise, we can see how understanding the initial position vector is crucial to solving for other elements like displacement.

The final position vector is where the object's journey ends after accounting for any displacement. If the journey of the positron starts from the initial position, the final position vector \( \vec{r} \) represents its stopping point.
This vector captures the endpoint and is equally as critical as the initial vector in understanding the object's trajectory. Like the initial position vector, the final position vector tells us where the object is in three-dimensional space, but it represents the destination instead.

• It is defined similarly to \( \vec{r}_0 \), with components along the x, y, and z-axes. For our positron, we have \( \vec{r} = 0 \hat{\mathrm{i}} + 3.0 \hat{\mathrm{j}} - 4.0 \hat{\mathrm{k}} \).
• This tells us the positron's position relative to the origin at the end of its journey.

Understanding these vectors allows us to chart the entire path the object takes from its initial to its final position.

Vector subtraction is a fundamental operation in physics and mathematics, crucial for understanding how two points are related in vector space. It helps determine changes between two vectors – such as positions. In our exercise, this is used to find the initial position vector when the final position is known.
The vector subtraction operation occurs component-wise, meaning each dimension of the vector is handled separately:

• For the \( \hat{\mathrm{i}} \) component, you subtract the initial from the final: \( 0 - 2.0 = -2.0 \hat{\mathrm{i}} \).
• For the \( \hat{\mathrm{j}} \) component, \( 3.0 - (-3.0) = 6.0 \hat{\mathrm{j}} \) involves adding instead of subtracting because of the double negative.
• Finally, \( \hat{\mathrm{k}} \) component involves the result of \( -4.0 - 6.0 = -10.0 \hat{\mathrm{k}} \).

This component-wise subtraction results in the overall vector, \( \vec{r}_0 = -2.0 \hat{\mathrm{i}} + 6.0 \hat{\mathrm{j}} - 10.0 \hat{\mathrm{k}} \), a result depicting the vector differences between the final and displaced vectors. Understanding vector subtraction in relation to initial and final positions enables us to solve problems involving trajectories and displacements effectively.