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how can a lose vertex "P" be made a part of selected line in "image 2". So that the mesh act as one entity? enter image description here

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Here is how I accomplish this when I am modeling:

First, select the edge you want the vertex to be added to. Then hit W > subdivide. This will add a new vertex on the edge, then select both the new vertex and then the original one (making sure to select them in that order) and hit Alt + M > at last. This will merge the two vertices to the location where the original vertex was.

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You can also accomplish this by using Ctrl+r loop cut and sliding it over to the point and using w -> Remove Doubles to 'weld' the two sections together. If the piece happens to be aligned with one of the axis (x,y,or z) it's very easy to get the alignment perfect by selecting the vertex and using Shift+s -> 'Cursor to Selected' and making sure that the Pivot Center is set to 3D Cursor as shown here. Then press s for scale and use z and 0 which will constrain the scaling operation to the z-axis and it will scale to zero. Since the pivot was set to 3D Cursor, it will scale to the point that was used to position the cursor.

enter image description here

I imagine you can also scale a line to a point by using local space/custom transform axis.

Here's an animation.

enter image description here

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