Magnitude of a Vector & Unit Vectors (DP IB Analysis & Approaches (AA): HL): Revision Note

Magnitude of a vector

What is the magnitude of a vector?

• The magnitude of a vector tells us its length

  • The magnitude of a displacement vector is the distance between the two points

  • The magnitude of a position vector is the distance of the point from the origin

• The magnitude of the vector is denoted by two vertical lines around the vector

  • For example, the magnitude of the vector a is denoted |a|

How do I find the magnitude of a vector?

• The magnitude of a vector can be found using Pythagoras’ theorem

• The formula for the magnitude of a vector is $\left|\mathit{v}\right|= \sqrt{v_1^2+\mathbf{ }{v}_2^{ 2}+v_3^{ 2}}$

  • $\mathit{v}= \left(\begin{array}{c}\genfrac{}{}{0pt}{}{v_1}{v_2}\\ v_3\end{array}\right)$

How do I find the distance between two points using vectors?

• STEP 1
  Find the displacement vector between them

• STEP 2
  Find the magnitude of the displacement vector

Unit vectors

What is a unit vector?

• A unit vector has a magnitude of 1

• You can find a unit vector that is in the same direction as the vector $\mathit{a}$ by dividing the components of the vector by the magnitude of the vector

  • $\frac{\mathit{a}}{\left|\mathit{a}\right|}$ is a unit vector in the direction of $\mathit{a}$

  • For example, a unit vector in the direction $3\mathit{i}–4\mathit{j}$ is $\frac{1}{\sqrt{3^2+4^2}}(3\mathit{i}-4\mathit{j})=\frac{3}{5}\mathit{i}-\frac{4}{5}\mathit{j}$