Components, resolved parts and the vectors i and j

Introduction to Vector Components

In your journey through Further Pure Mathematics, you have already seen that a vector represents a movement with both a magnitude (size) and a direction. But drawing arrows on paper every time you want to solve a problem is slow and sometimes inaccurate.

To make math easier, we use a coordinate-style system to describe vectors. This involves breaking a vector down into its components using two special building blocks: the vectors \(\mathbf{i}\) and \(\mathbf{j}\). Think of these as the "DNA" of every 2D vector!

The Unit Vectors \(\mathbf{i}\) and \(\mathbf{j}\)

Instead of saying "go five miles North-East," mathematicians prefer to say "go so many units right, and so many units up." To do this, we use unit vectors. A unit vector is simply a vector with a magnitude of \(1\).

In our 2D (coplanar) world, we define two standard unit vectors:

1. \(\mathbf{i}\): A vector of length \(1\) in the positive \(x\)-direction (to the right).
2. \(\mathbf{j}\): A vector of length \(1\) in the positive \(y\)-direction (straight up).

Handy Tip: In textbooks, these are printed in bold (\(\mathbf{i}\) and \(\mathbf{j}\)). When you are writing them by hand in your exam, you should underline them (\(\underline{i}\) and \(\underline{j}\)) to show they are vectors and not just regular numbers!

Quick Review:
\(\mathbf{i}\) = 1 step Right
\(-\mathbf{i}\) = 1 step Left
\(\mathbf{j}\) = 1 step Up
\(-\mathbf{j}\) = 1 step Down

Writing Vectors in Component Form

Any vector can be written as a combination of these two units. This is called component form. If a vector \(\mathbf{a}\) moves \(x\) units horizontally and \(y\) units vertically, we write it as:

\(\mathbf{a} = x\mathbf{i} + y\mathbf{j}\)

In this expression:
- \(x\) is the horizontal component.
- \(y\) is the vertical component.

Example: If a vector \(\mathbf{v}\) moves 4 units right and 3 units down, it is written as:
\(\mathbf{v} = 4\mathbf{i} - 3\mathbf{j}\)

Did you know?

This is very similar to coordinates \((x, y)\). The main difference is that a coordinate describes a fixed point, while a vector describes the movement or the gap between points.

Resolved Parts of a Vector

When we "resolve" a vector, we are essentially breaking it apart to see how much of it acts in a specific direction (usually the \(x\) and \(y\) directions).

If you are given a vector \(\mathbf{r} = 5\mathbf{i} + 2\mathbf{j}\):
- The resolved part in the direction of \(\mathbf{i}\) is \(5\).
- The resolved part in the direction of \(\mathbf{j}\) is \(2\).

Sometimes, a vector might only have one component. For example, the vector \(7\mathbf{i}\) is purely horizontal. Its resolved part in the \(\mathbf{j}\) direction is simply \(0\).

Why is this useful?

Don't worry if this seems like just another way of writing things—it actually makes calculations much simpler! For instance, if you want to add two vectors, you just add their components separately:

If \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{b} = 4\mathbf{i} - 1\mathbf{j}\), then:
\(\mathbf{a} + \mathbf{b} = (2 + 4)\mathbf{i} + (3 - 1)\mathbf{j} = 6\mathbf{i} + 2\mathbf{j}\)

Common Mistakes to Avoid

1. Mixing up \(\mathbf{i}\) and \(\mathbf{j}\): Always remember that \(\mathbf{i}\) comes first alphabetically and represents the horizontal \(x\)-axis, just like in coordinates \((x, y)\).
2. Ignoring Signs: If a vector moves left or down, the component must be negative. "3 units left" is \(-3\mathbf{i}\).
3. Adding \(\mathbf{i}\) to \(\mathbf{j}\): You cannot combine \(\mathbf{i}\) and \(\mathbf{j}\) into a single term. \(2\mathbf{i} + 3\mathbf{j}\) is NOT \(5\mathbf{ij}\). They are like apples and oranges!

Step-by-Step: Writing a Vector from a Diagram

If you are looking at a vector arrow on a grid and need to write it in \(\mathbf{i}, \mathbf{j}\) form:
1. Start at the "tail" of the arrow.
2. Count how many squares you move horizontally to get to the "head". (Right is positive, Left is negative). This is your \(\mathbf{i}\) coefficient.
3. Count how many squares you move vertically. (Up is positive, Down is negative). This is your \(\mathbf{j}\) coefficient.
4. Combine them: \(x\mathbf{i} + y\mathbf{j}\).

Summary Key Takeaways

- \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors (length of 1) in the \(x\) and \(y\) directions.
- Components are the values of \(x\) and \(y\) in the expression \(x\mathbf{i} + y\mathbf{j}\).
- Resolving a vector means finding these horizontal and vertical parts.
- This notation allows us to perform algebra on vectors instead of just drawing them.

Note: For information on finding the length of these vectors, see the next chapter on Magnitude, unit vectors and position vectors.