Magnitude of a Vector
In short: The magnitude of a vector is its length, found from the components with the Pythagorean theorem: for ⟨a, b⟩ it equals the square root of a² + b², always zero or positive.
PrecalculusGeometryN-VM.A.1
What is Magnitude of a Vector?
The magnitude of a vector is how long its arrow is, ignoring which way it points. A wind of 25 miles per hour from the west and a wind of 25 miles per hour from the east have different directions but the same magnitude.
For \(\mathbf{v} = \langle a, b\rangle\), the length is \(\|\mathbf{v}\| = \sqrt{a^2 + b^2}\), which comes straight from the Pythagorean theorem on the right triangle formed by the components. A vector of length 1 is a unit vector, and dividing any nonzero vector by its length produces one in the same direction.
How to spot it
- A question asks how fast, how strong or how far, which is the magnitude of a vector given by horizontal and vertical parts.
- Double bars or absolute-value bars surround a vector, as in the norm of v.
- You must turn a vector into a unit vector, which needs its length first.
- Two components such as 7 and 24 suggest a right triangle with a whole-number hypotenuse.
The method
- Write the components \(a\) and \(b\) of the vector.
- Square each component, so signs disappear, and add the squares.
- Take the positive square root of the sum; that root is the magnitude of a vector.
- For a unit vector, divide both components by this length.
Magnitude of a Vector: solved example
Problem
A gust is modeled by the vector \(\mathbf{w} = \langle -7, 24\rangle\) in miles per hour. Find its speed and the unit vector in the same direction.
Solution
Speed is the length: \(\|\mathbf{w}\| = \sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25\).
Divide each component by 25 for the unit vector: \(\left\langle -\frac{7}{25}, \frac{24}{25}\right\rangle\). The speed is 25 miles per hour.
Try it
Question 1
Find the length of \(\langle 9, -12\rangle\).
Show the answer
\(\sqrt{81 + 144} = \sqrt{225} = 15\).
Question 2
Find the length of \(\langle 2, 3\rangle\) to three decimal places.
Show the answer
\(\sqrt{4 + 9} = \sqrt{13} \approx 3.606\).
Magnitude of a Vector vs. Vector
The full arrow carries two facts: a heading and a size. Only the size is wanted here, and it is a single number that is never negative. Vectors \(\langle 5, 0\rangle\) and \(\langle 0, -5\rangle\) are different vectors but have equal length, 5.
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