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The parallelogram law is a fundamental principle for vector addition. It beautifully illustrates how two vectors can be added visually.
Here's the core idea:

• To use the parallelogram law, you place two vectors, say \(\overrightarrow{A}\) and \(\overrightarrow{B}\), so that they originate from the same point (tail to tail).
• Next, imagine each vector as a side of a parallelogram.
• The "resultant" vector, the sum of the two, is the diagonal running from the point where the vectors meet to the opposite corner of the parallelogram.

In our original problem, vectors \(\overrightarrow{OS}\) and \(\overrightarrow{SQ}\) are treated as sides of a parallelogram. Their sum, \(\overrightarrow{OQ} = (4, 6)\), represents the diagonal and thus confirms the solution. The parallelogram law is essential because it gives a quick visual depiction of vector addition, which is appreciated not only in physics but also in various fields of engineering and computer graphics.

The magnitude of a vector tells us about its size or length. It is crucial for understanding the spatial distance covered by the vector, irrespective of direction.
For any vector \((a, b)\), the magnitude is computed using the formula \[\sqrt{a^2 + b^2}\]Imagine you have a vector from point O (0, 0) to S (5, 9), represented as \(\overrightarrow{OS} = (5, 9)\). The magnitude would be:

• Calculate the squares: \(5^2 = 25\) and \(9^2 = 81\).
• Sum these values: \(25 + 81 = 106\).
• Take the square root: \(\sqrt{106}\).

This process shows how the length of the vector is essentially the hypotenuse of a right triangle with sides of length \(a\) and \(b\). Understanding the magnitude helps in various practical applications, such as determining velocity, where direction is secondary, and measuring distances in coordinate geometry.

Coordinate geometry, or analytic geometry, is where algebra meets geometry through graphs and coordinates. It offers an algebraic way of representing and analyzing spatial figures, crucial in performing tasks such as vector addition.
In this context:

• Any point in the plane can be represented as a coordinate pair, such as \((x, y)\).
• Vectors, then, are simply directed line segments from one point to another with both magnitude and direction.
• By defining vectors through coordinates, you can easily perform operations like addition or scaling.

Take the vector \(\overrightarrow{OS} = (5, 9)\) from our problem:

• \((5, 9)\) tells us that the vector starts at the origin \((0, 0)\) and points towards \(x = 5\), \(y = 9\).
• Changing these values will directly change the vector's orientation or starting/ending points.

Coordinate geometry simplifies solving various geometric problems, making it invaluable in both simple tasks and complex engineering design solutions.