Vector Operations and Column Notation

1. What is a Vector?

At its core, a vector is a quantity that has both magnitude (size or length) and direction. A scalar, by contrast, is just a number; it has magnitude but no direction (e.g., speed, mass, time).

You've already seen vectors represented as column vectors. The numbers in the column tell you the components of the movement.

• The top number is the horizontal movement (positive for right, negative for left).
• The bottom number is the vertical movement (positive for up, negative for down).

Let's start with a quick visual refresher on how column vectors represent movement on a grid.

Besides column vectors, you will also see vectors written as single bold lowercase letters (like a) or with an arrow if they connect two points (like $\overrightarrow{AB}$).

2. Vector Arithmetic: The Rules

Now, let's establish the rules for adding, subtracting, and multiplying vectors. The good news is that the arithmetic is very straightforward when using column notation. You simply perform the operations on the corresponding components.

To summarize the rules:

• Addition: Add the top components and add the bottom components.
  $\left(\begin{array}{c}a\\ b\end{array}\right)+\left(\begin{array}{c}c\\ d\end{array}\right)=\left(\begin{array}{c}a+c\\ b+d\end{array}\right)$
• Subtraction: Subtract the top components and subtract the bottom components.
  $\left(\begin{array}{c}a\\ b\end{array}\right)-\left(\begin{array}{c}c\\ d\end{array}\right)=\left(\begin{array}{c}a-c\\ b-d\end{array}\right)$
• Scalar Multiplication: Multiply both components by the scalar (the number).
  $k\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}ka\\ kb\end{array}\right)$

Let's watch these rules in action.

3. Applying Vector Arithmetic to Solve Problems

Performing the basic arithmetic is just the first step. The real power of vectors in your exam is in using them to solve geometric problems.

Finding the Vector Between Two Points

A common task is to find the vector that describes the journey from one point to another. If you have the position vectors of two points A and B (i.e., the vectors from the origin O to each point, $\overrightarrow{OA}$ and $\overrightarrow{OB}$), you can find the vector $\overrightarrow{AB}$ using a simple subtraction rule.

The Rule: $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$

A simple way to remember this is "end point minus start point".

Finding Vector Paths in Geometric Shapes

Vector arithmetic allows you to find a path from one point to another by adding up a "route" of known vectors. Think of it like a journey: if you go from A to B, and then B to C, the total journey is from A to C. So, $\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}$.

If you need to travel along a vector in the opposite direction, you subtract it. For example, $\overrightarrow{BA}=-\overrightarrow{AB}$.

Let's see this applied to a typical exam problem involving a parallelogram.

These problems often involve ratios and midpoints. The principle is the same: break the journey down into known parts. For example, if M is the midpoint of AB, then $\overrightarrow{AM}=\frac{1}{2}\overrightarrow{AB}$.

Independent Practice

To solidify your skills, it's essential to work through some exam-style questions.

Conclusion

Today you've built a solid foundation in vector arithmetic. This is a topic where precision is key, so always double-check your signs when adding and subtracting.

Key Takeaways:

• A vector has both magnitude and direction, and can be represented as a column vector $\left(\begin{array}{c}x\\ y\end{array}\right)$.
• Vector Addition/Subtraction: Perform the operation on the corresponding components.
• Scalar Multiplication: Multiply both components by the scalar.
• The vector from point A to point B can be found using the position vectors: $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$.
• You can find unknown vectors in geometric figures by creating a path using known vectors. Travelling against a vector's direction means you subtract it.

Preview of the Next Lesson:
We've defined a vector as having both direction and magnitude. In this lesson, we focused on direction and arithmetic. In the next lesson, we will focus on the other part: calculating the magnitude of a vector. You will learn how to find the length of a vector using a method that connects directly to Pythagoras' theorem.