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It is important to realize that magnetic lines do not really exist! They are a tool to visualize the magnetic field, but the field is continuous and does not exist solely inside lines. The direction of the lines gives the direction of the magnetic field, the density of lines, its strength.

This also explains why no two field lines can ever intersect; a field line carries information about the direction of the magnetic field, if they would intersect an ambiguity would arise about the direction (not to mention a field of apparent infinite strength since the density would be infinite at the point of crossing).

The field lines are almost never used in explicit calculations; instead one uses a vector, an entity which contains information about the magnitude and direction of a field in every point in space and time. Adding two magnetic fields is then easy; just add the vectors of both fields in every point in space (and time). You can use the resulting vector field to draw field lines again if you want.

An easy way to imagine what would happen to field lines when they might intersect is to look at them as being such vectors. Imagine you have one field line pointing to the right, and another one pointing up. The result of adding would be a field line pointing somewhere in the up-right direction (the exact direction depending on the relative magnitudes of the fields).

If the fields are equal in magnitude but opposite in direction they would cancel; the field line disappears. But this is to be expected! The magnetic fields canceled each other in that point!

One has to take care with this analogy however; as for field lines the measure of magnitude is their density; which is an undefined thing if you are considering just one field line per field. For a vector however, the measure of magnitude is its length. Therefore adding two field lines of the same magnitude and pointing in the same direction would result in a vector of twice the length, but in field line language you would have to double the density at that point.

This is one of the reasons field lines are used for visualization but not calculation.

By the way, all these things apply to other fields as well. Electric fields can also be represented by field lines, and they as well cannot intersect (for the same reasons). Electric field lines, however, are not necessarily closed loops like magnetic field lines (this has to do with the non-existence of magnetic monopoles).

Q: Why do two magnetic lines never intersect each other?

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No, horizontal lines are parallel to each other and parallel lines never intersect.

the parallel lines never intersect each other but they both intersect the line they are perpendicular to

The answer would be parallel lines these lines never meet or cross each other.

Parallel lines remain the same distance apart and never intersect each other whereas other types of lines intersect each other at some point.

Parallel lines never intersect and remain equal distance from each other

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No, horizontal lines are parallel to each other and parallel lines never intersect.

Perpendicular lines always intersect and make 90 degree angles. Parallel lines never intersect with each other.

Parallel lines are always the same distance from each other and never intersect.

the parallel lines never intersect each other but they both intersect the line they are perpendicular to

Parallel lines remain equal distance apart and never intersect each other.

They are asymptote lines in which as a curve gets closer and closer to them they will never intersect with each other.

The answer would be parallel lines these lines never meet or cross each other.

Parallel lines remain the same distance apart and never intersect each other whereas other types of lines intersect each other at some point.

Lines that never intersect are either parallel or skew to each other. If they're both in the same plane (or on the same piece of paper), then they're parallel.

Parallel lines never intersect and remain equal distance from each other

Because, parallel lines never intersect.

because they are solid lines and they do not have space to be combining. Also the magnetic can stay together but not melt to being a diffusion (not liquid)