91Ó°ÊÓ

Displacement vectors are used to represent movement from one position to another in a particular direction. These vectors not only have a magnitude, which represents the distance, but also a direction, pointing from the starting location to the end location.
In physics, and especially in this particular problem, displacement vectors are very helpful in simplifying complex problems to simple linear movements along a specified path.

• Magnitude: This tells us how far an object has moved.
• Direction: Gives us the path of movement, such as north or east.

In our exercise, we worked with two displacement vectors, \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathbf{B}}\), both pointing due north. Each vector indicates a specific movement along the north-south line. The task was to find the resultant displacement when we subtract these vectors from one another.

When dealing with vectors, finding the magnitude and direction of the resultant vector is crucial. The magnitude of a vector is found by calculating the scalar length of the vector. The direction describes where the vector points relative to a reference direction (e.g., north, south).
For the problem at hand, the magnitude and direction are determined by subtracting the respective vector components, as both vectors are aligned:

• Resultant Magnitude: Subtract the smaller magnitude from the larger one.
• Direction of Resultant: Assign the direction based on the sign of the result. A positive result leads to a northward direction, while a negative result reflects a southward direction.

In step 2 of our solution, \(\overrightarrow{\mathrm{A}} - \overrightarrow{\mathrm{B}}\) gives us a magnitude of -5.31 km, indicating a southward direction, while \(\overrightarrow{\mathbf{B}} - \overrightarrow{\mathrm{A}}\) results in +5.31 km, pointing north.

Collinear vectors refer to vectors that lie along the same line. This means they either point in the same direction or directly opposite directions. Understandably, their manipulation simplifies into mere algebraic addition or subtraction, primarily because their directional components align perfectly.
For vectors like \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathbf{B}}\), both moving due north, they are collinear.
This establishes some simplifying rules:

• Additive Nature: When vectors point similarly, their magnitudes add up.
• Subtractive Nature: For opposite directions, magnitudes subtract, revealing a difference indicating either a reverse direction or a lesser magnitude.

In our task, since \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathbf{B}}\) align, their subtraction was straightforward, resulting in either a positive (northward) or negative (southward) direction, based solely on the subtraction sequence.