What is the Magnitude of a Vector?
A vector is a quantity that has both direction and size (magnitude). The magnitude of a vector is the length of the line segment representing it, regardless of its direction. For example, if vector AB starts at point A and ends at point B, the magnitude is the distance from A to B.
The magnitude is denoted by vertical bars around the vector symbol, such as |AB|, which reads as 'the magnitude of vector AB.' This notation emphasises that we are interested in the size, not the direction, of the vector.
- Magnitude represents the length of the vector.
- Denoted by vertical bars: |AB| means the size of vector AB.
- Magnitude is always zero or positive.
Using Coordinates to Find Vector Magnitude
When points A and B are given in a coordinate plane, their coordinates allow us to calculate the magnitude of vector AB precisely. Suppose A has coordinates (x₁, y₁) and B has coordinates (x₂, y₂). The vector AB points from A to B.
The magnitude |AB| is the straight-line distance between these two points. This distance can be found using the distance formula derived from the Pythagorean theorem.
- Identify coordinates of points A and B.
- Use the distance formula to find |AB|.
- Magnitude is the length of the segment AB.
The Distance Formula and Calculation
The distance formula between two points (x₁, y₁) and (x₂, y₂) is:
\[ |AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula calculates the length of the hypotenuse of a right-angled triangle formed by the horizontal and vertical differences between the points.
To calculate the magnitude, subtract the x-coordinates and y-coordinates, square these differences, add them, and then take the square root of the sum.
- Calculate horizontal difference: (x₂ - x₁).
- Calculate vertical difference: (y₂ - y₁).
- Square both differences and add.
- Take the square root to find |AB|.
Worked Example
Find the magnitude of vector AB where A is at (2, 3) and B is at (7, 11).
Step 1: Calculate the differences: x₂ - x₁ = 7 - 2 = 5, y₂ - y₁ = 11 - 3 = 8.
Step 2: Square the differences: 5² = 25, 8² = 64, sum = 25 + 64 = 89, Step 3: Take the square root: \( \sqrt{89} \approx 9.43 \). Therefore, |AB| ≈ 9.43 units.
- Use coordinates (2, 3) and (7, 11).
- Calculate differences: 5 and 8.
- Sum of squares: 89.
- Magnitude: approximately 9.43 units.
Key Points for Exam Success
Always write the magnitude with vertical bars, for example |AB|, to show you are calculating the size of the vector.
Show all steps clearly: state coordinates, calculate differences, square, sum, and square root.
Remember magnitude cannot be negative; if you get a negative value, check your calculations carefully as the square root is always non-negative in this context.
- Use correct notation: |AB| for magnitude.
- Show full calculation steps.
- Magnitude is always ≥ 0.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify points | Find coordinates of vector start and end points | A(2,3), B(7,11) | Always label points clearly |
| Calculate differences | Subtract coordinates: x₂ - x₁ and y₂ - y₁ | 5 and 8 | Write differences explicitly |
| Square and add | Square differences and sum: 5² + 8² = 25 + 64 = 89 | Sum is 89 | Show intermediate sums |
| Square root | Take √89 ≈ 9.43 | Magnitude ≈ 9.43 units | Final answer with units |
Active recall
Quick check
1. If A is at (1, 2) and B is at (4, 6), what is |AB|?
Calculate differences: 4 - 1 = 3, 6 - 2 = 4; square and add: 3² + 4² = 9 + 16 = 25; square root: √25 = 5; so |AB| = 5.
2. What does the notation |AB| represent?
It represents the magnitude (length) of vector AB, the distance between points A and B.
3. Can the magnitude of a vector be negative?
No, magnitude is always zero or positive because it represents length.