﻿ Formula for the Length of a Vector

# Formula for the Length of a Vector

by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
• PRACTICE (online exercises and printable worksheets)
• You may want to review prior sections:

Let $\,\vec v = \langle a,b\rangle\,$ be a vector.

Depending upon the signs (plus or minus) of $\,a\,$ and $\,b\,$,
the vector $\,\vec v\,$ is one of the four vectors shown below.

(To match the diagram, suppose that $\,a\,$ and $\,b\,$ are both nonzero.)

 In all four cases, the length (size, magnitude) of $\,\vec v\,$ is the hypotenuse of a triangle with sides of length $\,|a|\,$ and $\,|b|\,$. Recall that $\,\|\vec v\|\,$ denotes the length of $\,\vec v\,$. We have: \begin{alignat}{2} \cssId{s7}{\|\vec v\|^2} \quad &=\quad \cssId{s8}{|a|^2 + |b|^2} &&\qquad \cssId{s9}{\text{(by the Pythagorean Theorem)}}\cr &=\quad \cssId{s10}{a^2 + b^2} &&\qquad \cssId{s11}{\text{(x^2 = |x|^2\,, since they have the same size and sign)}}\cr \end{alignat} Take the square root of both sides, and use the fact that $\,\|\vec v\|\ge 0\,$. The result is the formula for the length of $\,\vec v = \langle a,b\rangle\,$: $$\|\vec v\| = \sqrt{a^2 + b^2} \qquad \text{(vector length formula)}$$ $\vec v = \langle a,b\rangle\,$ is one of these four vectors: $$\|\vec v\| = \sqrt{a^2 + b^2}$$

## Notes on the Vector Length Formula

• In words, to find the length of a vector:
• square the horizontal component
• square the vertical component
• take the square root of the sum
More compactly:
The length of a vector is the square root of the sum of the squares of the horizontal and vertical components.
• If the horizontal or vertical component is zero:
If $\,a\,$ or $\,b\,$ is zero, then you don't need the vector length formula.
In this case, the length is just the absolute value of the nonzero component.
For example:
• $\,\|\,\langle 5,0\rangle\,\| = 5\,$
• $\,\|\,\langle 0,-3\rangle\,\| = |-3| = 3\,$
In general:
• $\,\|\,\langle a,0\rangle\,\| = |a|\,$
• $\,\|\,\langle 0,b\rangle\,\| = |b|\,$
However, the vector length formula certainly works—keep reading!
• The correct formula for taking the square root of a square:
For all real numbers $\,x\,$, $\sqrt{x^2} = |x|\,$.
Without the absolute value symbol, it doesn't work for negative numbers!
For example, $\,\sqrt{(-3)^2} \ne -3\,$. Be careful!
• Using the Vector Length Formula when a component is zero:
Here's what happens with the vector length formula if one of the components is zero:
$$\cssId{s42}{\|\,\langle a,0\rangle\,\|} = \cssId{s43}{\sqrt{a^2 + 0^2}} = \cssId{s44}{\sqrt{a^2}} = \cssId{s45}{|a|} \qquad\qquad \cssId{s46}{\text{ and }} \qquad\qquad \cssId{s47}{\|\,\langle 0,b\rangle\,\|} = \cssId{s48}{\sqrt{0^2 + b^2}} = \cssId{s49}{\sqrt{b^2}} = \cssId{s50}{|b|}$$ The formula works, but it's unnecessary in these simple cases.

## Finding the Length of a Scaled Vector

Let $\,a\,$, $\,b\,$, and $\,k\,$ be real numbers.
Let $\,\vec v = \langle a,b\rangle\,$.
Then, \begin{alignat}{2} \cssId{s56}{\|k\vec v\|} \quad&=\quad \cssId{s57}{\|\,k\langle a,b\rangle\,\|} \qquad\qquad&&\cssId{s58}{\text{(definition of \,\vec v\,)}}\cr &=\quad \cssId{s59}{\|\,\langle ka,kb \rangle\,\|} \qquad\qquad&&\cssId{s60}{\text{(multiply a vector by a scalar)}}\cr &=\quad \cssId{s61}{\sqrt{(ka)^2 + (kb)^2}} \qquad\qquad&&\cssId{s62}{\text{(the vector length formula)}}\cr &=\quad \cssId{s63}{\sqrt{k^2a^2 + k^2b^2}} \qquad\qquad&&\cssId{s64}{\text{(exponent law, squaring a product)}}\cr &=\quad \cssId{s65}{\sqrt{k^2(a^2 + b^2)} } \qquad\qquad&&\cssId{s66}{\text{(factor)}}\cr &=\quad \cssId{s67}{\sqrt{k^2}\sqrt{a^2 + b^2}} \qquad\qquad&&\cssId{s68}{\text{(property of radicals)}}\cr &=\quad \cssId{s69}{|k| \cdot \sqrt{a^2 + b^2}} \qquad\qquad&&\cssId{s70}{\text{(\ \sqrt{x^2}=|x|\ )}}\cr &=\quad \cssId{s71}{|k|\cdot \|\vec v\| } \qquad\qquad&&\cssId{s72}{\text{(the vector length formula)}} \end{alignat} So, $\,\|k\vec v\| = |k|\cdot \|\vec v\|\,$.

Master the ideas from this section

When you're done practicing, move on to:
vectors: from direction/magnitude
to horizontal/vertical components

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