   Chapter 13.2, Problem 54E

Chapter
Section
Textbook Problem

# Find an expression for d d t [ u ( t ) ⋅ ( v ( t ) × w ( t ) ) ] .

To determine

To find: An expression for ddt[u(t)(v(t)×w(t))] .

Explanation

Formula used:

Write the expression to find the derivative of dot product of the two vectors u(t) and v(t) .

ddt[u(t)v(t)]=u(t)v(t)+u(t)v(t) (1)

Write the expression to find the derivative of cross product of the two vectors u(t) and v(t) .

ddt[u(t)×v(t)]=u(t)×v(t)+u(t)×v(t) (2)

Determination of expression for ddt[u(t)(v(t)×w(t))] :

Substitute u(t) for u(t) and [v(t)×w(t)] for v(t) in equation (1).

ddt{u(t)[v(t)×w(t)]}=u(t)[v(t)×w(t)]+u(t)ddt[v(t)×w(t)] (3)

Calculation of vector ddt[v(t)×w(t)] :

Substitute v(t) for u(t) and w(t) for v(t) in equation (2),

ddt[v(t)×w(t)]=v(t)×w(t)+v(t)×w(t)

Substitute [v(t)×w(t)+v(t)×w(t)] for ddt[v(t)×w(t)] in equation (3),

ddt{u(t)[v(t)×w(t)]}={u(t)[v(t)×w(t)]+u(t)[v(t)×w(t)+v(t)×w(t)]}={u(t)[v(t)×w(t)]+u(t)[v(t)×w(t)]+u(t)[v(t)×w(t)]}

Use the following properties of cross product and dot product, and compute the expression

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