Orthogonality · Definition

Vector norm

linear algebraorthogonality

Before you start you need to know

It gives meaning to «how long» a vector is and lets us talk about distances and unit-length vectors in any dimension.

Definition

The norm of a vector in \(\mathbb{R}^n\) is a nonnegative number measuring its length: the square root of the vector's dot product with itself.

\[\|x\|=\sqrt{x\cdot x}=\sqrt{x_1^2+\dots+x_n^2}\]

The norm obeys three basic rules: \(\|v\|\ge0\), with equality only for the zero vector; \(\|cv\|=|c|\,\|v\|\), so scaling stretches or shrinks the length; and the triangle inequality \(\|u+v\|\le\|u\|+\|v\|\), which says that going straight is never longer than taking a detour. Equality holds only when the vectors point the same way.

Computing a norm is applying the Pythagorean theorem in several coordinates. For instance, \(\|(1,2,2)\|=\sqrt{1+4+4}=3\). Dividing a nonzero vector by its norm gives a unit vector in the same direction: \(\tfrac{1}{5}(3,-4)\) is the unit vector of \((3,-4)\). The distance between points \(x\) and \(y\) is \(\|x-y\|\).

Expanding \(\|u+v\|^2=\|u\|^2+2\,u\cdot v+\|v\|^2\) produces the cross term \(2\,u\cdot v\), which vanishes exactly when \(u\cdot v=0\); this leads to orthogonality and the Pythagorean theorem in \(\mathbb{R}^n\). A typical mistake is forgetting the root and giving \(x\cdot x\) as the norm, or assuming \(\|u+v\|=\|u\|+\|v\|\) always.

Think about it

When does \(\|u+v\|=\|u\|+\|v\|\) hold?

When \(u\) and \(v\) point the same way (one is a nonnegative multiple of the other); otherwise the triangle inequality is strict.

What is the norm of \(\tfrac{v}{\|v\|}\) if \(v\neq0\)?

One: \(\left\|\tfrac{v}{\|v\|}\right\|=\tfrac{1}{\|v\|}\|v\|=1\).

Exercises 10

  1. Compute \(\|(3,4)\|\).
  2. Compute \(\|(1,2,2)\|\).
  3. Compute \(\|(2,-3,6)\|\).
  4. If \(\|v\|=4\), what is \(\|3v\|\)?
  5. How do \(\|v\|\) and \(\|-v\|\) compare?
  6. Which unit vector points in the direction of \((3,-4)\)?
  7. Find \(k>0\) such that \(\|(k,2k,2k)\|=12\).
  8. If \(\|u\|=3\) and \(\|v\|=4\), which of these values is impossible for \(\|u+v\|\)?
  9. If \(\|u\|=2\), \(\|v\|=3\) and \(u\cdot v=-1\), what is \(\|u+v\|\)?
  10. Which statement holds for all vectors \(u,v\) and every real \(c\)?

What comes next

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §1.2
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §6.A