AcademyMeasurement and Vectors
Academy
Unit Vector Notation
Level 1 - Physics topic page in Measurement and Vectors.
Principle
Unit-vector notation writes signed components along chosen basis directions.
Notation
\(\hat{\imath}\)
positive x direction
none
\(\hat{\jmath}\)
positive y direction
none
\(\hat{k}\)
positive z direction
none
\(a_x\)
x-component
same as vector
\(a_y\)
y-component
same as vector
\(a_z\)
z-component
same as vector
Method
The basis vectors show direction only; the component coefficients carry the physical units.
Like basis directions combine; unlike basis directions stay separate.
Vector form
\[\vec{a}=a_x\hat{\imath}+a_y\hat{\jmath}+a_z\hat{k}\]
Magnitude
\[|\vec{a}|=\sqrt{a_x^2+a_y^2+a_z^2}\]
Like terms
\[a_x\hat{\imath}+b_x\hat{\imath}=(a_x+b_x)\hat{\imath}\]
Rules
Vector form
\[\vec{a}=a_x\hat{\imath}+a_y\hat{\jmath}+a_z\hat{k}\]
Vector magnitude
\[|\vec{a}|=\sqrt{a_x^2+a_y^2+a_z^2}\]
Add vectors
\[\vec{a}+\vec{b}=(a_x+b_x)\hat{\imath}+(a_y+b_y)\hat{\jmath}+(a_z+b_z)\hat{k}\]
Checks
- Unit vectors have no physical units.
- Components carry physical units.
- Combine only matching basis directions.
- A negative component points opposite its basis vector.