Welcome to Basic Operations on Vectors!

In this chapter, we are moving beyond simple numbers (scalars) and diving into the world of vectors. While a scalar is just a value (like your age or the temperature), a vector tells us two things: how much and in what direction. Think of it like a set of instructions: "Walk 5 miles" is a scalar, but "Walk 5 miles North" is a vector.

Don't worry if this seems a bit abstract at first. We’re going to look at how to add them, subtract them, and stretch them using simple steps that work every single time!


1. Vector Addition: Joining the Path

Adding vectors is like following a map. If vector \(\mathbf{a}\) tells you how to get from point A to point B, and vector \(\mathbf{b}\) tells you how to get from B to C, then \(\mathbf{a} + \mathbf{b}\) is the shortcut that takes you straight from A to C.

The Algebraic Way (The "Easy" Way)

When vectors are written as column vectors or in \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) notation, addition is as simple as adding the corresponding numbers together.

If \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\), then:
\(\mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}\)

Example: If \(\mathbf{a} = 3\mathbf{i} + 2\mathbf{j}\) and \(\mathbf{b} = 1\mathbf{i} - 5\mathbf{j}\):
\(\mathbf{a} + \mathbf{b} = (3+1)\mathbf{i} + (2-5)\mathbf{j} = 4\mathbf{i} - 3\mathbf{j}\)

The Geometric Way (The Triangle Law)

To add vectors visually, we use the Tip-to-Tail method:
1. Draw the first vector (\(\mathbf{a}\)).
2. Draw the second vector (\(\mathbf{b}\)) starting from the tip (the arrow end) of the first one.
3. The resultant vector (\(\mathbf{a} + \mathbf{b}\)) is the line drawn from the very start to the very end.

Quick Review: Adding vectors is just "combining instructions." Just add the top numbers together and the bottom numbers together!


2. Scalar Multiplication: Stretching and Flipping

A scalar is just a normal number (like 2, 0.5, or -3). When we multiply a vector by a scalar, we are scaling it.

What happens to the vector?

  • If you multiply by a number greater than 1, the vector gets longer.
  • If you multiply by a number between 0 and 1, the vector gets shorter.
  • If you multiply by a negative number, the vector reverses direction.

Mathematically, you just multiply every component of the vector by that number:
\(k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}\)

Example: If \(\mathbf{v} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}\), then \(3\mathbf{v} = \begin{pmatrix} 12 \\ -6 \end{pmatrix}\).
The new vector is three times as long but still points in the same direction.

Did you know? Two vectors are parallel if one is a scalar multiple of the other. For example, \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) and \(\begin{pmatrix} 5 \\ 10 \end{pmatrix}\) are parallel because the second one is just \(5 \times\) the first!


3. Vector Subtraction: Finding the Difference

Subtracting a vector is exactly the same as adding a negative vector.
\(\mathbf{a} - \mathbf{b}\) is the same as \(\mathbf{a} + (-\mathbf{b})\).

How to do it:

1. Algebraically: Subtract the components. (Top minus top, bottom minus bottom).
2. Geometrically: To draw \(-\mathbf{b}\), just take vector \(\mathbf{b}\) and flip the arrow head to the other end. Then, add it to \(\mathbf{a}\) using the tip-to-tail method.

Memory Aid: Think of \(\mathbf{a} - \mathbf{b}\) as the vector that takes you from the tip of b to the tip of a when they both start at the same point.


4. Working in Three Dimensions (3D)

Good news! Everything you learned about 2D vectors works exactly the same for 3D vectors. You just have one extra number to deal with (the \(z\) component, or \(\mathbf{k}\)).

If \(\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}\), then:
\(\mathbf{a} + \mathbf{b} = \begin{pmatrix} 1+4 \\ 2+5 \\ 3+6 \end{pmatrix} = \begin{pmatrix} 5 \\ 7 \\ 9 \end{pmatrix}\)

Key Takeaway: Don't let the extra dimension scare you. Whether it's addition, subtraction, or scaling, you just treat each row (\(x, y, z\)) as its own little math problem.


5. Common Mistakes to Avoid

  • Mixing up Scalars and Vectors: Never add a plain number to a vector. You can’t do \(5 + \begin{pmatrix} 1 \\ 2 \end{pmatrix}\). It’s like trying to add "5" to "North"—it doesn't make sense!
  • Handwriting Notation: In exams, you can't write in bold. You must use an underline (e.g., \(\underline{u}\)) to show a letter is a vector. This is a very common way to lose easy marks!
  • Subtraction Direction: When drawing \(\mathbf{a} - \mathbf{b}\), students often point the arrow the wrong way. Remember: it points towards the vector you are starting with (\(\mathbf{a}\)).

Chapter Summary

Checklist for Success:

1. Addition: Use tip-to-tail for diagrams; add components for algebra.
2. Scaling: Multiply all components by the scalar; negative scalars reverse the direction.
3. Parallel Vectors: Look for multiples (e.g., \(\mathbf{b} = k\mathbf{a}\)).
4. Notation: Always underline your vectors when writing by hand!