Key takeaway: The magnitude of a vector is its length, found by applying the Pythagorean theorem to its components, so ⟨a, b⟩ has magnitude √(a² + b²).
Grade 12 – PrecalculusN-VM.A.1Geometry and Measurement
Definition of Magnitude of a Vector
The magnitude of a vector is the answer to “how big?” It ignores the heading entirely and reports only the length of the arrow. For a velocity vector the magnitude is the speed; for a force vector it is the strength of the push.
Because the horizontal and vertical components form the legs of a right triangle, with the arrow itself as the hypotenuse, the length follows from the Pythagorean theorem. Dividing a nonzero arrow by its length produces a unit vector, which has magnitude exactly 1 and keeps the original direction. The magnitude is always zero or positive and equals zero only for the zero arrow.
Formula and notation
The magnitude is written with bars, \(\|\mathbf{v}\|\) or \(|\mathbf{v}|\):
\[\|\langle a,b\rangle\|=\sqrt{a^2+b^2},\qquad \hat{\mathbf{v}}=\frac{\mathbf{v}}{\|\mathbf{v}\|}.\]
The second expression is the unit vector in the direction of \(\mathbf{v}\).
Key facts
- Multiplying every component by k scales the magnitude of a vector by |k|.
- Opposite arrows have identical lengths, so the sign of components does not matter.
- In three dimensions a third squared component is added under the root.
- The length of a sum is at most the sum of the lengths of its parts.
Magnitude of a Vector example
A drone’s velocity is \(\mathbf{v}=\langle 7,4\rangle\) in miles per hour (east, north). Find its speed and the unit vector in its direction of travel.
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Square and add the components: \(7^2+4^2=49+16=65\).
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Take the root: speed \(=\sqrt{65}\approx 8.06\) mph.
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Divide each component by that length: \(\hat{\mathbf{v}}=\langle\frac{7}{\sqrt{65}},\frac{4}{\sqrt{65}}\rangle\approx\langle 0.868,0.496\rangle\).
The speed is \(\sqrt{65}\approx 8.06\) mph, along the unit direction \(\langle 0.868,0.496\rangle\).
Common mistake
Watch out
Hector adds the components and reports a speed of \(7+4=11\) mph.
Why it is wrong
Adding the legs of a right triangle does not give the hypotenuse; walking 7 east then 4 north is longer than the straight-line path. To get the magnitude of a vector, square, add, then take the root: about 8.06 mph.
Related terms and lessons
Browse the glossary: A–Z list · Precalculus vocabulary · Geometry and Measurement