Magnitude of a Two-Dimensional Vector
Simple Explanation
The magnitude (length) of a column vector v = (x, y) is found using the Pythagorean theorem: |v| = √(x² + y²).
Why Do We Need It?
This converts a vector's abstract component form back into a concrete, measurable length — essential whenever an actual real-world size (like a distance or speed) is needed from column vector data.
See It
An arrow from the origin to the point (4,3), with dashed horizontal and vertical component segments of length 4 and 3 forming a right triangle
Formula
Magnitude of a Column Vector
|v| = √(x² + y²), for v = (x, y)
The magnitude (length) of a two-dimensional vector is found by applying the Pythagorean theorem to its horizontal and vertical components.
- v
- — a two-dimensional vector, written in column/component form
- x, y
- — the horizontal and vertical components of v
- |v|
- — the magnitude (length) of v
When to use it: Whenever a vector is given in component (column) form and you need its actual length.
Worked Example
Find the magnitude of a column vector
Find the magnitude of the vector v = (4, 3).
Why Does This Work?
The vector's x and y components form the two legs of a right triangle, since they are perpendicular (horizontal versus vertical). The vector itself — the straight-line arrow from the origin to (x, y) — is exactly the hypotenuse of that triangle, so its length follows directly from the Pythagorean theorem.
Real-Life Example
A hiker's straight-line distance from camp
A hiker walks 4 km east, then 3 km north, from their campsite.
Even though 7 km were actually walked, the hiker's direct (straight-line) distance from camp is the magnitude of the resultant vector (4,3), which is only 5 km.
Practice
Find the magnitude of the vector w = (6, 8).
MediumCommon mistake
Adding the components directly (x+y) instead of squaring each, summing, and taking the square root — magnitude is almost never just x+y.
Quick Review
- |v| = √(x² + y²), from the Pythagorean theorem.
- x and y are the legs of a right triangle; |v| is the hypotenuse.
- v=(4,3) → |v|=5 and v=(6,8) → |v|=10 are useful reference triangles.