You can use the following expression - Either as a Geometry Generator symbol layer (on Layer B) or by using the Geometry by Expression geoprocessing tool if you wish to create a new layer with the results (used also on Layer B).
-- find the intersection of the line layers and assign it to a variable
with_variable('i',
intersection(
-- a small extension is applied to Layer A to ensure there is a true intersection with Layer B.
-- the extension is applied at both ends so that the direction of the line does not need to be known
aggregate('Layer A', 'collect', extend(@geometry, 1, 1)),
@geometry
),
-- get the distance of the intersection along Layer B's geometry
with_variable('dist_along',
line_locate_point(@geometry, @i),
-- generate the line that spans 100 m either side of the intersection
line_substring(@geometry, @dist_along - 100, @dist_along + 100)
)
)
Note, when using either a Geometry Generator symbol layer or the Geometry by Expression tool, don't forget to set the output geometry type to LineString / MultiLineString
or Line
respectively.