As per MySQL manual:

If the table has a multiple-column index, any leftmost prefix of the index can be used by the optimizer to find rows. For example, if you have a three-column index on (col1, col2, col3), you have indexed search capabilities on (col1), (col1, col2), and (col1, col2, col3).

MySQL cannot use an index if the columns do not form a leftmost prefix of the index. Suppose that you have the SELECT statements shown here:

SELECT * FROM tbl_name WHERE col1=val1;
SELECT * FROM tbl_name WHERE col1=val1 AND col2=val2;

SELECT * FROM tbl_name WHERE col2=val2;
SELECT * FROM tbl_name WHERE col2=val2 AND col3=val3;

If an index exists on (col1, col2, col3), only the first two queries use the index. The third and fourth queries do involve indexed columns, but (col2) and (col2, col3) are not leftmost prefixes of (col1, col2, col3).
Now, my question is how I should define the indexes if I want to execute the following queries efficiently:

Code:
SELECT * FROM tbl_name WHERE col1=val1 AND col2=val2;
SELECT * FROM tbl_name WHERE col1=val1 AND col3=val3;
SELECT * FROM tbl_name WHERE col1=val1 AND col2=val2 AND col3=val3;
An index on (col1, col2, col3) doesn't work with the 2nd query. If instead I would define an index on (col1, col2) and (col1, col3) would that help the 3rd query?