   Chapter 12.2, Problem 77E

Chapter
Section
Textbook Problem

# Perpendicular Vectors Consider the vector-valued function r ( t ) = ( e t sin t ) i + ( e t cos t ) j . Show that r(t) and r " ( t ) are always perpendicular to each other.

To determine

To prove: The vector functions r(t) and r(t ) are perpendicular to the vector-valued function r(t)=(etsint)i+(etcost)j.

Explanation

Given:

The vector-valued function is r(t)=(etsint)i+(etcost)j

Formula used:

The product formula is:

ddtuv=uv'+vu'

Proof:

The provided expression is,

r(t)=(etsint)i+(etcost)j

Differentiate with respect to t,

r(t)=(etsint+etcost )i+(etcostetsint)j

Again differentiate,

r(t)=(etsint+etcost+etcostetsint)i+(etcostetsintetsintetcost)j=(2etcost)i(2etsint)j

Now, two vectors are perpendicular if their scalar product is zero

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