91Ó°ÊÓ

Coordinate geometry is a branch of mathematics that helps us deal with geometry using a coordinate system. A coordinate system uses numbers as coordinates to describe the position of points on a plane.

In our exercise, we work in a two-dimensional plane with pairs of numbers, \(x, y\), representing each point like \((x, y)\). For example, point \(P(-1, 3)\) tells us that \(-1\) is how far 'left or right' it is from the origin (0,0), and \(3\) is how far 'up or down'.

• The negative sign with \(-1\) means it is to the left of the origin.
• The positive \(3\) means it is above the origin.

Understanding the position of each point in coordinate geometry is fundamental to determining characteristics like distance or direction.

The magnitude of a vector tells us about its length. It's a bit like figuring out how long a stick would measure if it was positioned between two points. In math, we find the magnitude by using the Pythagorean theorem.
For a vector with components \(x\) and \(y\), its magnitude is represented by \(\|\mathbf{v}\|\) and calculated as: \[\|\mathbf{v}\| = \sqrt{x^2 + y^2}\].

In our problem, the resultant vector \(\overrightarrow{OP} + \overrightarrow{PQ}\) has the coordinates \(4,6\). To find its magnitude, we plug \(4\) and \(6\) into our formula:
\[\|\mathbf{v}\| = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} .\]
This magnitude tells us the straight-line distance from point \(O\) to the resultant point in the plane. It's an important property in physics and engineering where vectors represent quantities like velocity or force.

The parallelogram law is a visual way to understand vector addition. Imagine drawing a parallelogram where two vectors serve as adjacent sides. The diagonal of this parallelogram represents the sum of the two vectors.

In our specific case, \(\overrightarrow{OP}\) and \(\overrightarrow{PQ}\) create adjacent sides. The diagonal from \(O\) represents \(\overrightarrow{OP} + \overrightarrow{PQ}\). This diagonal or resultant vector should match our earlier calculated sum.

• Start at \(O\), draw \(\overrightarrow{OP}\).
• Then from \(P\), draw \(\overrightarrow{PQ}\).
• Complete the parallelogram from these points.
• The vector from \(O\) to the opposite corner of the parallelogram is \(\overrightarrow{OA}\).

By applying the parallelogram law, we confirm both visually and mathematically that \(\overrightarrow{OA} = (4, 6)\), providing an intuitive check.

Vector components break a vector down into parts based on the coordinate directions. Think of a vector as having two fingers pointing: one horizontally along the x-axis, and the other vertically along the y-axis.

• The x-component tells how far along the x-direction the vector is.
• The y-component shows how far along the y-direction.

In our problem, vector \(\overrightarrow{OP}\) is made up of components \(-1\) (for the x-direction) and \(3\) (for the y-direction). When adding vectors, it's helpful to sum their respective components separately, like we did with \(\overrightarrow{OP}\) and \(\overrightarrow{PQ}\).

For example, the sum vector \(\overrightarrow{OP} + \overrightarrow{PQ} = (4, 6)\) was calculated by adding the x-components together and the y-components together: \((-1 + 5, 3 + 3) = (4, 6)\). This method makes the math straightforward and logical while following the rules of coordinate geometry thoroughly.