Vector Derivatives
How to Differentiate a Vector
The derivative of a vector-valued function is found by differentiating each of its components individually: $$ \frac{d \ \vec{v(t)}}{dt} = \frac{d \ v_x(t)}{dt} \vec{u_x} + \frac{d \ v_y(t)}{dt} \vec{u_y} + \frac{d \ v_z(t)}{dt} \vec{u_z} $$
Here, \( \vec{u_x} \), \( \vec{u_y} \), and \( \vec{u_z} \) denote the unit vectors of the reference frame, typically corresponding to the Cartesian axes x, y, and z.
These unit vectors are often alternatively represented by \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).
Note. Differentiating a vector follows the same rules and properties of differentiation that apply to real-valued functions.
A Worked Example
Consider the following vector-valued function, expressed as the sum of its component functions:
$$ \vec{v(t)} = [\sin(\pi t)]\, \vec{u_x} + [2t^2]\, \vec{u_y} + [\cos(\pi t)]\, \vec{u_z} $$
The derivative of \( \vec{v}(t) \) with respect to time \( t \) is obtained by differentiating each component separately:
$$ \frac{d \ \vec{v(t)}}{dt} = \left[ \frac{d}{dt} \sin(\pi t) \right] \vec{u_x} + \left[ \frac{d}{dt} 2t^2 \right] \vec{u_y} + \left[ \frac{d}{dt} \cos(\pi t) \right] \vec{u_z} $$
Computing each derivative individually gives:
$$ \frac{d \ \vec{v(t)}}{dt} = [\pi \cos(\pi t)]\, \vec{u_x} + [4t]\, \vec{u_y} - [\pi \sin(\pi t)]\, \vec{u_z} $$
This expression represents the time derivative of the vector.
Derivative of the Dot Product
To differentiate the dot product of two vector-valued functions, we apply the product rule: $$ \frac{d}{dt}[\vec{a} \cdot \vec{b}] = \frac{d \vec{a}}{dt} \cdot \vec{b} + \vec{a} \cdot \frac{d \vec{b}}{dt} $$
The derivative of the dot product is the sum of two dot products:
- The dot product of the derivative of \( \vec{a} \) with \( \vec{b} \) (undifferentiated)
- The dot product of \( \vec{a} \) with the derivative of \( \vec{b} \)
Derivative of the Cross Product
Similarly, to differentiate the cross product of two vectors, we apply the product rule adapted to the cross product: $$ \frac{d}{dt}[\vec{a} \times \vec{b}] = \frac{d \vec{a}}{dt} \times \vec{b} + \vec{a} \times \frac{d \vec{b}}{dt} $$
The derivative of the cross product is the sum of two cross products:
- The cross product of the derivative of \( \vec{a} \) with \( \vec{b} \)
- The cross product of \( \vec{a} \) with the derivative of \( \vec{b} \)
And so forth.
