Vectors & Polar Coordinates

Vector Operations

Adding, subtracting, and scaling vectors in component form.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

A Number That Points Somewhere

Before reading on, think about the difference between saying a hiker walked '5 miles' and saying they walked '5 miles northeast.' The first is just a distance — a plain number. What does the second sentence capture that the first one completely leaves out?

Definition — Vector

A quantity with both magnitude (size) and direction, written in component form as ⟨a, b⟩ — a horizontal and a vertical displacement. A plain number with no direction, like an ordinary distance, is called a scalar by contrast. A vector's magnitude is |v| = √(a² + b²).

That magnitude formula is exactly the distance formula (equivalently, the Pythagorean Theorem) applied to a vector's own components — a vector's horizontal and vertical parts form the two legs of a right triangle, and its magnitude is the resulting hypotenuse.

Vectors add by combining their components separately — and this matches a genuine geometric picture: placing the second vector's tail at the first vector's tip (tip-to-tail), the resultant vector's own components are exactly the total horizontal and total vertical displacement from start to finish, which is precisely what adding each component separately computes.

Worked Example — Adding and Subtracting Vectors

Let v = ⟨3, 4⟩ and w = ⟨−1, 2⟩. Find v + w and v − w. Adding: ⟨3 + (−1), 4 + 2⟩ = ⟨2, 6⟩. Subtracting: ⟨3 − (−1), 4 − 2⟩ = ⟨4, 2⟩.

Worked Example — Finding Magnitude and Scaling a Vector

Find the magnitude of v = ⟨3, 4⟩, then find 2v and its magnitude. Magnitude: |v| = √(9 + 16) = √25 = 5. Scaling: 2v = ⟨6, 8⟩, with magnitude √(36 + 64) = √100 = 10 — exactly double the original, confirming that scaling a vector by a factor scales its magnitude by that same factor.

Tip

Scaling a vector by a positive number stretches or shrinks it without changing its direction; scaling by a negative number also reverses its direction — a quick way to predict a scaled vector's general appearance before computing its exact components.

Common Mistakes

Treating a vector as if it were just its magnitude, discarding the direction information entirely.
A vector's identity depends on both its magnitude and direction — two vectors with the same magnitude but different directions are genuinely different vectors, not interchangeable.
Making a sign error when subtracting vector components, especially when the second vector has negative components.
Subtracting a vector's component means subtracting that exact value, sign included — v − w = ⟨3 − (−1), ...⟩ becomes ⟨3 + 1, ...⟩, not ⟨3 − 1, ...⟩.

Key Takeaways

A vector has both magnitude and direction, unlike a scalar, which is just a plain number.
A vector's magnitude, √(a² + b²), comes directly from the Pythagorean Theorem applied to its components.
Vector addition combines components separately, matching the geometric tip-to-tail picture of combining displacements.

Summary

Vectors formalize quantities with both size and direction, using components and the same distance-formula reasoning from coordinate geometry. The next lesson introduces an alternative coordinate system built directly around magnitude and direction.

Track your progress

Back to Vectors & Polar Coordinates Rectangular to Polar Conversion→