Question

Show that if r is a vector function such that exists, then \(\frac{d}{{dt}}(r(t) \times r'(t)) = r(t) \times {r^n}(t)\).

Short answer

Hence proved \(\frac{d}{{dt}}(r(t) \times r'(t)) = r(t) \times r''(t)\).

Step-by-step solution

Step 1: Product Rule

Using product rule we can write

\(\begin{array}{l}\frac{d}{{dt}}(r(t) \times r'(t)) = \frac{d}{{dt}}(r(t) \times r'(t) + r(t) \times \frac{d}{{dt}}(r'(t))\\\frac{d}{{dt}}(r(t) \times r'(t)) = r'(t) \times r'(t) + r(t) \times r''(t)\end{array}\)

Step 2: The conclusion

\(\frac{d}{{dt}}(r(t) \times r'(t)) = r'(t) \times r'(t) + r(t) \times r''(t)\)

We know that cross product of vector with itself is zero

Hence proved that

\(\frac{d}{{dt}}(r(t) \times r'(t)) = r(t) \times r''(t)\)