Question

Let $r\left(t\right)$be a vector-valued function whose graph is a curve C, and let $a\left(t\right)$be the acceleration vector. Prove that if $a\left(t\right)$is always zero, then C is a straight line.

Short answer

Ans: If $a\left(t\right)=⟨0,0,0⟩$then the graph is a straight line.

Step-by-step solution

Step 1. Given information:

The graph of the vectors valued function $r\left(t\right)$is on curve C. $a\left(t\right)$be the acceleration vector.

Step 2. Proving C is a straight line:

Let $a\left(t\right)=⟨0,0,0⟩$

The velocity vector, $v\left(t\right)=\int a\left(t\right)dt$

$=\int ⟨0,0,0⟩dt$

$=\left(c_{1}t+\alpha \right)i+\left(c_{2}t+\beta \right)j+\left(c_{3}t+\gamma \right)k$where $\alpha ,\beta ,\gamma$are constants.

, the graph is a straight line because it is in linear form.

Thus if the acceleration vector of$r\left(t\right)$is always zero, then the graph of$r\left(t\right),c$, is a straight line.