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In physics, when we talk about vector representation, we are looking into how we can express quantities that have both magnitude and direction. Velocity is one such quantity. Cars moving towards a corner can be perfectly represented using vectors. This is helpful because vectors help us understand not just how fast something is moving, but also in which direction.
In the given exercise, each car's velocity is represented as a vector:

• Car 1's velocity vector is expressed in the form \( \mathbf{v}_1 = 35 \hat{i} \) km/h. Here \( \hat{i} \) indicates the direction along the x-axis (say eastward).
• Car 2's velocity vector is \( \mathbf{v}_2 = 45 \hat{j} \) km/h, where \( \hat{j} \) points towards the y-axis (say northward).

These vectors are not just lines with arrows, but mathematical representations that allow us to perform operations like addition and subtraction easily. In relative velocity problems, such operations become crucial as they let us find out velocities of objects as observed from one another.

A coordinate system is essentially an arrangement that helps you plot or define positions and directions. Imagine it like a map where you need to know which way is north or east to navigate. In physics, we assign directions to the axes commonly known as x (horizontal) and y (vertical). This gives us a clear reference point.
When solving the exercise, placing Car 1's movement along the x-axis and Car 2's along the y-axis simplifies calculations.

• Car 1's movement in positive x-direction corresponds to its velocity vector being placed along the x-axis.
• Similarly, Car 2's motion is strictly along the y-direction.

This perpendicular alignment is what helps apply vector math effectively, as we can respect both direction and magnitude when doing calculations. As a result, it's easier to find the relative velocities by just subtracting these vectors.

The Pythagorean theorem is a principle from geometry that helps us relate the sides of right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. It's usually noted as: \[ c^2 = a^2 + b^2 \].
In the context of our exercise, the relative velocity vector of one car as seen from the other forms a right-angled triangle with the individual velocities. To find the magnitude of this relative velocity:

• The formula applied is \[ \|\mathbf{v}_{1,2}\| = \sqrt{(35)^2 + (-45)^2} \], which provides \(57.00\) km/h as calculated.
• This same formula helps find the magnitude of the relative velocity regardless of the perspective since the main difference is the direction, not the magnitude.

Therefore, the Pythagorean theorem becomes a simple yet powerful tool for transforming vector components into tangible velocities in real-world problems.