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 $\,\vec v = \langle a,b\rangle\,$:
$$
\|\vec v\| = \sqrt{a^2 + b^2} \qquad \text{(vector length formula)}
$$
|
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
- add these squares together
- take the square root of the sum
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:- $\,\|\,\langle 5,0\rangle\,\| = 5\,$
- $\,\|\,\langle 0,-3\rangle\,\| = |-3| = 3\,$
- $\,\|\,\langle a,0\rangle\,\| = |a|\,$
- $\,\|\,\langle 0,b\rangle\,\| = |b|\,$
-
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, $\,\|k\vec v\| = |k|\cdot \|\vec v\|\,.$
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
vectors: from direction/magnitude
to horizontal/vertical components