Components of Vectors (DP IB Applications & Interpretation (AI): HL): Revision Note

Components of vectors

What are components of a vector?

• Components of a vector describe how the vector acts in different directions

  • Consider $4\mathit{i}-3\mathit{j}+\mathit{k}$ as an example

    • The component in the $\mathit{i}$ direction is 4

    • The component in the $\mathit{j}$ direction is -3

• You need to know how to find the component of a vector that is:

  • Parallel to another vector

  • Perpendicular to another vector

How do I find the component of a vector that is parallel to another vector?

• Suppose you want to find the component of $\mathit{a}$ that is acting in the direction of $\mathit{b}$

• If you know the angle between $\mathit{a}$ and $\mathit{b}$

  • The component of $\mathit{a}$ acting parallel to $\mathit{b}$ is $\left|\mathit{a}\right|\cos \theta$

• You can also find the component using the scalar product

  • The formula is $\frac{\mathit{a}·\mathit{b}}{\left|\mathit{b}\right|}$

How do I find the component of a vector that is perpendicular to another vector?

• Suppose you want to find the component of $\mathit{a}$ that is acting perpendicular to $\mathit{b}$

  • There are an infinite number of directions that are perpendicular to a vector in 3D

  • Therefore, you want to find the component that is in the plane formed by the two vectors

• If you know the angle between $\mathit{a}$ and $\mathit{b}$

  • The component of $\mathit{a}$ acting perpendicular to $\mathit{b}$ is $\left|\mathit{a}\right|\sin \theta$

• You can also find the component using the vector product

  • The formula is $\frac{\left|\mathit{a}\times \mathit{b}\right|}{\left|\mathit{b}\right|}$