Introduction to Vector Arithmetic
Welcome to the world of vector arithmetic! In our previous look at vectors, we learned that a vector is a quantity that has both magnitude (size) and direction. In this chapter, we are going to learn how to do math with them. Don't worry if it sounds complicated—at its heart, vector arithmetic is just like basic addition and subtraction, just with two numbers at once!
Think of a vector as a set of instructions: "Walk 3 steps East and 4 steps North." Vector arithmetic is simply combining these sets of instructions to see where you end up. This is a vital skill for navigation, physics, and computer graphics.
Note: For the basics of what a vector is and how to write it, check out the "Vectors and vector notation" chapter.
1. Adding and Subtracting Vectors
When we add or subtract vectors, we are finding the "total" movement. There are two ways to look at this: using column vectors and using diagrams.
Using Column Vectors
This is the easiest way to perform arithmetic. If you have two vectors, you simply add or subtract the "top" numbers (x-components) and the "bottom" numbers (y-components) separately.
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 \(\mathbf{a} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 1 \\ -5 \end{pmatrix}\):
\(\mathbf{a} + \mathbf{b} = \begin{pmatrix} 3 + 1 \\ 2 + (-5) \end{pmatrix} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\)
Using Diagrams (The Tip-to-Tail Method)
To add vectors \(\mathbf{a}\) and \(\mathbf{b}\) on a diagram, follow these steps:
1. Draw vector \(\mathbf{a}\).
2. Start drawing vector \(\mathbf{b}\) from the end (the arrow tip) of vector \(\mathbf{a}\).
3. The result is the vector that goes from the very start of \(\mathbf{a}\) to the very end of \(\mathbf{b}\).
Subtraction: To subtract \(\mathbf{b}\), you just add the negative of \(\mathbf{b}\). If \(\mathbf{b}\) goes right and up, \(-\mathbf{b}\) goes left and down with the exact same steepness.
Quick Review:
- Add: Top + Top, Bottom + Bottom.
- Subtract: Top - Top, Bottom - Bottom.
- Geometrically: Follow the arrows!
2. Multiplication by a Scalar
In vector math, a scalar is just a normal number (like 2, 5, or -0.5). When you multiply a vector by a scalar, you are changing its size, and sometimes its direction.
To multiply a vector by a scalar \(k\), multiply every number inside the column vector by \(k\):
\(k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}\)
What does this do visually?
- If \(k = 2\), the vector becomes twice as long but stays in the same direction.
- If \(k = 0.5\), the vector becomes half as long.
- If \(k = -1\), the vector stays the same length but points in the opposite direction.
Parallel Vectors: If one vector is a scalar multiple of another (e.g., \(\mathbf{a} = 3\mathbf{b}\)), then the vectors are parallel. They are pointed in the same (or exactly opposite) direction!
3. Magnitude (Modulus) of a Vector
The magnitude of a vector is simply its length. In your exam, the notation for the magnitude of vector \(\mathbf{a}\) is \(|\mathbf{a}|\).
Because a vector is made of a horizontal move (\(x\)) and a vertical move (\(y\)), they form a right-angled triangle. This means we can use Pythagoras' Theorem to find the length!
For any vector \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\), the magnitude is calculated as:
\(|\mathbf{a}| = \sqrt{x^2 + y^2}\)
Example: Find the magnitude of \(\mathbf{v} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}\).
\(|\mathbf{v}| = \sqrt{3^2 + (-4)^2}\)
\(|\mathbf{v}| = \sqrt{9 + 16}\)
\(|\mathbf{v}| = \sqrt{25} = 5\)
Did you know? A unit vector is any vector with a magnitude of exactly 1. It is often used to show a pure direction without worrying about distance.
4. The Resultant Vector
The resultant is just a fancy word for the "final" vector you get after adding two or more vectors together. It represents the single direct path from the start point to the final end point.
If a plane flies with vector \(\mathbf{p}\) but the wind blows it with vector \(\mathbf{w}\), the actual path of the plane is the resultant vector \(\mathbf{r} = \mathbf{p} + \mathbf{w}\).
Step-by-Step for Resultant Problems:
1. Identify all the individual vectors given.
2. Write them as column vectors if they aren't already.
3. Add them all together: \(\mathbf{Resultant} = \mathbf{a} + \mathbf{b} + \mathbf{c} + ...\)
4. If the question asks for the size of the resultant, calculate its magnitude using \(\sqrt{x^2 + y^2}\).
Common Mistakes to Avoid
1. Squaring Negative Numbers: When calculating magnitude, remember that \((-4)^2\) is positive \(16\), not \(-16\). Magnitudes are always positive lengths!
2. Mixing up X and Y: Always remember: Top is Right/Left (x), Bottom is Up/Down (y). Don't swap them!
3. Forgetting Units: If the question gives units like cm or m, make sure your final magnitude answer includes them.
4. Notation Errors: Remember that \(\mathbf{a}\) (bold) or \(\vec{AB}\) means the vector itself, but \(|\mathbf{a}|\) or \(AB\) (no arrow) refers to the length only.
Key Takeaways
- Addition/Subtraction: Perform the operation on the top and bottom numbers independently.
- Scalar Multiplication: Multiply both components by the number; this creates a parallel vector.
- Magnitude: Use Pythagoras: \(|\mathbf{a}| = \sqrt{x^2 + y^2}\).
- Resultant: The sum of two or more vectors.
- Direction: If a scalar is negative, the direction of the vector reverses.
Don't worry if this seems tricky at first! Practicing with column vectors is usually very successful for students because it works just like the addition you already know. Once you master the "Top and Bottom" rule, you've mastered 90% of vector arithmetic!