Grabbing the rope and swinging on the vine in Tomb Raider 4
Are you also lost in how to use the rope to jump from one platform to the other? Yea, I had the same problem too.
All you have to do is grab the rope with the Action button (CTRL by default) while you are mid-air, then use the Dash button (by default, it should be the semicolon ";"), the same you use for sprinting while running, to swing on the rope. You can use arrow keys to move up and down. The lower you are on the rope, the better you can swing.
You can also turn Lara in order to change her swinging direction by pressing Left and Right arrows.
To jump off, just let go of the Action button (CTRL) or jump by using ALT.

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Here is a screenshot of the controls I use:

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