Magnitude of a Vector (SQA National 5 Maths): Revision Note

Calculating the magnitude of a vector

What is the magnitude of a vector?

• The magnitude of a vector is its length

  • It is also called the modulus

  • This is always a positive value

  • The direction of the vector is irrelevant

• The magnitude of vector $\overrightarrow{\mathit{A}\mathit{B}}$ is written $\left|\overrightarrow{AB}\right|$

  • The magnitude of vector a is written |a|

• In real world contexts, the magnitude of a vector can represent different quantities

  • For a velocity vector, its magnitude would be speed

  • For a force vector, its magnitude would be the strength of the force (in newtons)

How do I find the magnitude of a two-dimensional vector?

• For a 2D vector in component form, the magnitude is the hypotenuse of a right-angled triangle

  • Use Pythagoras' theorem to find the magnitude

  • The magnitude of $\mathbf{a}=\left(\begin{array}{c}x\\ y\end{array}\right)$ is

    • $\left|\mathbf{a}\right|=\sqrt{x^2+\mathbf{ }{y}^2}$

How do I find the magnitude of a three-dimensional vector?

• For a 3D vector in component form, the magnitude can be found using the following formula

  • The magnitude of $\mathbf{a}=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$ is

    • $\left|\mathbf{a}\right|=\sqrt{x^2+\mathbf{ }{y}^2+z^2}$

• This is formula is related to Pythagoras' Theorem in 3D