Formula for the Length of a Vector

• 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 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

• 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:
$$\|\,\langle a,0\rangle\,\| = \sqrt{a^2 + 0^2} = \sqrt{a^2} = |a| \qquad\qquad \text{ and } \qquad\qquad \|\,\langle 0,b\rangle\,\| = \sqrt{0^2 + b^2} = \sqrt{b^2} = |b|$$ 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} \|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