Answer
See the detailed answer below.
Work Step by Step
We need to find the work done by the force during these 3 different paths to see if this force is a conservative force or not.
We know that the work is given by
$$W=\vec F\;\vec r=(6\;\hat i+8\;\hat j)\vec r\tag 1$$
- For the first path:
$$W_{\rm ABD}=W_{\rm AB}+W_{\rm BD}$$
Plugging from (1);
$$W_{\rm ABD}=(6\;\hat i+8\;\hat j)\vec r_{\rm AB}+(6\;\hat i+8\;\hat j)\vec r_{\rm BD}$$
$$W_{\rm ABD}=(6\;\hat i+8\;\hat j)(3\;\hat i+0\;\hat j)+(6\;\hat i+8\;\hat j)(0\;\hat i+4\;\hat j) $$
Recall that $\hat i\cdot \hat i=1$ and that $\hat i\cdot \hat j=0$
$$W_{\rm ABD}= 18 +32 =\color{red}{\bf 50}\;\rm J$$
- For the second path:
$$W_{\rm ACD}=W_{\rm AC}+W_{\rm CD}$$
$$W_{\rm ACD}=(6\;\hat i+8\;\hat j)\vec r_{\rm AC}+(6\;\hat i+8\;\hat j)\vec r_{\rm CD}$$
$$W_{\rm ACD}=(6\;\hat i+8\;\hat j)(0\;\hat i+4\;\hat j)+(6\;\hat i+8\;\hat j)(3\;\hat i+0\;\hat j)$$
$$W_{\rm ABD}= 32+18=\color{red}{\bf 50}\;\rm J$$
- For the third path:
$$W_{\rm AD}=(6\;\hat i+8\;\hat j)\vec r_{\rm AD} =(6\;\hat i+8\;\hat j) (3\;\hat i+4\;\hat j) $$
$$W_{\rm AD}= 18+32=\color{red}{\bf 50}\;\rm J$$
And since the work through the three paths is the same, then this force is a conservative force.
