Formula for the Length of a Vector

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 at right.
(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: $$ \begin{alignat}{2} \|\vec v\|^2 \quad &=\quad |a|^2 + |b|^2 &&\qquad \text{(by the Pythagorean Theorem)}\cr &=\quad a^2 + b^2 &&\qquad \text{($x^2 = |x|^2\,$, since they have the same size and sign)}\cr \end{alignat} $$ 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

Finding the Length of a Scaled Vector

Let $\,a\,$, $\,b\,$, and $\,k\,$ be real numbers.
Let $\,\vec v = \langle a,b\rangle\,$.
Then, $$ \begin{alignat}{2} \|k\vec v\| \quad&=\quad \|\,k\langle a,b\rangle\,\| \qquad\qquad&&\text{(definition of $\,\vec v\,$)}\cr &=\quad \|\,\langle ka,kb \rangle\,\| \qquad\qquad&&\text{(multiply a vector by a scalar)}\cr &=\quad \sqrt{(ka)^2 + (kb)^2} \qquad\qquad&&\text{(the vector length formula)}\cr &=\quad \sqrt{k^2a^2 + k^2b^2} \qquad\qquad&&\text{(exponent law, squaring a product)}\cr &=\quad \sqrt{k^2(a^2 + b^2)} \qquad\qquad&&\text{(factor)}\cr &=\quad \sqrt{k^2}\sqrt{a^2 + b^2} \qquad\qquad&&\text{(property of radicals)}\cr &=\quad |k| \cdot \sqrt{a^2 + b^2} \qquad\qquad&&\text{($\ \sqrt{x^2}=|x|\ $)}\cr &=\quad |k|\cdot \|\vec v\| \qquad\qquad&&\text{(the vector length formula)} \end{alignat} $$ So, $\,\|k\vec v\| = |k|\cdot \|\vec v\|\,$.

Master the ideas from this section
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


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