It sound like instead of the intersect you may be using the merge. The three items are then what is known as a multipart objects. You can use the explode on multiparts to get the singleparts. However If you use the intersect command correctly it should give what you want.
Be sure to select the Line layer first. Then select the Polygon layer. (that could be the other way to get what your results are).
ESRI Desktop Help:
If you are trying to intersect line features, they must share a common
edge. For example, two lines that cross one another (for example, as
an X) share a common point but not a common edge. Use the Split Tool
on the 3D Editor toolbar to break this type of line intersection.
The Intersect command works on selected features, allowing you to
choose which features are intersected in the output, although you
cannot specify the output's attributes. On the other hand, the
Intersect geoprocessing tool creates new features from the overlaps in
all input layers or feature classes, and it allows you to determine
which attributes from the input features are transferred to the output
Steps: Click the Edit Placement button on the 3D Editor toolbar.
Click the features from whose intersection you want to create a new
feature. Click 3D Editor and click Intersect from the drop-down menu.
Choose the target in which the new feature will be created. If you
have feature templates for the layers in your 3D view, click the
Template button and click the template in which to create the new
feature. You can also double-click the preview of the template to
choose a different template. If you do not have feature templates,
click the layer in which to create the new feature. Click OK.
Related Topics Creating 3D feature data Duplicating existing features
vertically Combining features from different layers (Union) Merging
features in the same layer Copying parallel line features