# 虎克定律矢量形式

（\$ | v_ {1} ^ *-v_ {2} ^ * | \$：剩余長度）

（\$ | v_ {1}-v_ {2} | \$：當前長度）

\$ F_s =（\ frac {k_s | v_1 -v_2 |-| v_ {1} ^ *-v_ {2} ^ * |} {{| v_ {1} ^ *-v_ {2} ^ * |}}）\ frac {（v_1 -v_2）} {| v_1 -v_2 |} \$ //我認為\$ k_s \$是錯字。

\$ F_s = -k_s（\ frac {| v_1 -v_2 |-| v_ {1} ^ *-v_ {2} ^ * |} {{| v_ {1} ^ *-v_ {2} ^ *|}}）\ frac {（v_1 -v_2）} {| v_1 -v_2 |} \\\ quad = -k_s（\ frac {|（5,0,0）-（0,0,0）|--|（3,0,0）-（0,0,0）|} {{|（3,0,0）-（0,0,0）|}}}）\ frac {（（5,0,0）-（0,0,0））} {|（5,0,0）-（0,0,0）|} \\\ quad =-\ frac {2} {3} k_s（1,0,0）\$

\$-\ frac {2} {3} k_s（1,0,0）\$的作用力與\$ -2k_s \$完全不同

If we sidestep your typo (the last term has one absolute too much), both formulations are correct. They just express different things. The \$k\$ in Hooke's law is for a particular spring. \$k_s\$ is the siffness for a paricular material.

Now in the linear portion there is a direct relationship betwen these the material stffness is directly propotional to the spring factor and length of member. So for example when the rest length of a different spring with same material is 2 times as long its 1/2 as stiff.