Vectors in two dimensions

Welcome to the World of Vectors!

In this chapter, we are moving beyond simple numbers. Usually, in Math, we deal with scalars—things like mass or temperature that only have a "size." But in the real world, direction matters! If you are told a treasure is 5km away, you won't find it unless you know which direction to walk. That is exactly what a vector is: a quantity that has both magnitude (size) and direction.

Vectors are used by pilots to navigate planes through wind, by game developers to move characters on a screen, and by engineers to build bridges. Let's dive in!

Did you know? The word "vector" comes from Latin, meaning "carrier." It literally carries you from one point to another!

1. What does a Vector look like?

There are two main ways we write and see vectors in O-Level Math:

A. Directed Line Segments

Think of a vector as an arrow. The length of the arrow shows the size (magnitude), and the arrowhead shows the direction. We name them in two ways:
1. Using two capital letters with an arrow on top: \(\vec{AB}\) (this means the path starts at point \(A\) and ends at point \(B\)).
2. Using a single bold or underlined small letter: a or \(\underline{a}\).

B. Column Vector Notation

This is the most common way to write vectors in exams. We write it as:
\(\binom{x}{y}\)
- The top number \(x\) tells you how many units to move horizontally (Right is positive, Left is negative).
- The bottom number \(y\) tells you how many units to move vertically (Up is positive, Down is negative).

Example: If \(\vec{a} = \binom{3}{-2}\), it means you move 3 units to the right and 2 units down.

Quick Tip: Don't confuse this with coordinates \((x, y)\). A coordinate is a fixed "spot," but a vector is a "movement" or a "shift."

Key Takeaway: Vectors tell us "how far" and "which way."

2. Finding the Magnitude (Length)

The magnitude of a vector is simply how long the arrow is. We use vertical bars to represent this: \(|\vec{AB}|\) or \(|\mathbf{a}|\).

To find the magnitude of \(\binom{x}{y}\), we use our old friend, Pythagoras' Theorem!

\(|\binom{x}{y}| = \sqrt{x^2 + y^2}\)

Example: Find the magnitude of \(\vec{v} = \binom{3}{4}\).
\(|\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\) units.

Don't worry if this seems tricky: Just remember that the vector forms a right-angled triangle, and you are finding the longest side (the hypotenuse).

3. Scalar Multiplication

A scalar is just a normal number. When you multiply a vector by a scalar, you are "scaling" it (making it longer, shorter, or reversing its direction).

If \(k\) is a number, then \(k\binom{x}{y} = \binom{kx}{ky}\).

Example: If \(\mathbf{a} = \binom{2}{5}\), then \(3\mathbf{a} = \binom{3 \times 2}{3 \times 5} = \binom{6}{15}\).
The new vector \(3\mathbf{a}\) is 3 times as long as \(\mathbf{a}\) and points in the same direction.

Wait! What if the number is negative?
If you multiply by \(-1\), the vector stays the same length but points in the opposite direction.
If \(\vec{AB} = \mathbf{a}\), then \(\vec{BA} = -\mathbf{a}\).

Key Takeaway: Multiplying by a number changes the length. A negative sign flips the direction.

4. Adding and Subtracting Vectors

Imagine walking from home to the bus stop (\(\vec{u}\)), and then from the bus stop to school (\(\vec{v}\)). The total journey is \(\vec{u} + \vec{v}\).

Addition (The "Head-to-Tail" Rule)

To add vectors, just add the top numbers and add the bottom numbers:
\(\binom{x_1}{y_1} + \binom{x_2}{y_2} = \binom{x_1 + x_2}{y_1 + y_2}\)

Subtraction

Subtracting is just as easy:
\(\binom{x_1}{y_1} - \binom{x_2}{y_2} = \binom{x_1 - x_2}{y_1 - y_2}\)

Analogy: Think of vector addition as following a map. "Go 2 blocks East, then 3 blocks North." The resultant vector is the "shortcut" from your start to your finish.

5. Position Vectors

A Position Vector is a special vector that always starts from the Origin \(O\) (0,0).
If a point \(P\) has coordinates \((4, 7)\), its position vector is \(\vec{OP} = \binom{4}{7}\).

Important Formula for Exams:
To find the vector between any two points \(A\) and \(B\):
\(\vec{AB} = \vec{OB} - \vec{OA}\)
(Memory Aid: "Destination minus Starting point" or "Backwards subtraction")

6. Vectors in Geometry (Solving Problems)

Exam questions often ask you to express paths in terms of given vectors like \(\mathbf{a}\) and \(\mathbf{b}\). Here are the golden rules:

Rule 1: Parallel Vectors

If two vectors are parallel, one is just a multiple of the other.
If \(\mathbf{u} = k\mathbf{v}\), then \(\mathbf{u}\) is parallel to \(\mathbf{v}\).

Rule 2: Collinear Points (Points on a straight line)

If points \(A\), \(B\), and \(C\) lie on a straight line:
1. \(\vec{AB}\) must be parallel to \(\vec{BC}\) (meaning \(\vec{AB} = k\vec{BC}\)).
2. They must share a common point (like \(B\)).

Rule 3: Midpoints

If \(M\) is the midpoint of \(AB\), then \(\vec{AM} = \frac{1}{2}\vec{AB}\).

7. Common Mistakes to Avoid

1. Mixing up \(x\) and \(y\): Always remember \(x\) is horizontal (left/right) and \(y\) is vertical (up/down).
2. Forgetting the minus sign: If you go from \(B\) to \(A\) instead of \(A\) to \(B\), you must change the sign.
3. Magnitude math: When calculating \(\sqrt{x^2 + y^2}\), remember that a negative number squared (like \((-3)^2\)) becomes positive (9). Magnitude can never be negative!

Quick Review Box

Column Vector: \(\binom{right/left}{up/down}\)
Magnitude: \(\sqrt{x^2 + y^2}\)
Opposite Vector: \(\vec{BA} = -\vec{AB}\)
Path Rule: \(\vec{AC} = \vec{AB} + \vec{BC}\)
Parallel: \(\vec{a} = k\vec{b}\)

Congratulations! You've just covered the essentials of Vectors. Keep practicing drawing the paths with your pen—it makes a huge difference!