# MATH 125- Final Exam Guide - Comprehensive Notes for the exam ( 349 pages long!)

## Document Summary

Equations of lines and planes a summary. ~x = ~p + t ~d x = p1 + td1 y = p2 + td2. ~x = ~p + t ~d x = p1 + td1 y = p2 + td2 z = p3 + td3. Notes: some steps are skipped in the solutions below (e. g. , some row reduction of matrices is omitted). In your exam, you should show all your work! Remember also that there are often a number of ways to solve a given problem; here i have generally just given one method: (9 points) compute the determinant of q, where. = 1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (expand along 1st column) (r1 + r2) (expand along 1st column) 5 (c) the reduced row echelon form of the augmented matrix of the homogeneous system a~x = ~0 is equal to [b|~0], i. e. ,