Logs, complex numbers, matrices and vectors are the toolkit behind circuits, vibrations, statics and dynamics. Learn them here once, then use them everywhere.
Choose any section below to get a free practice problem. Work it with the FE Reference Handbook open, tap your answer, then check the solution.
A logarithm answers one question: what power do I raise the base to?
log10(0.001) equals:
0.001 = 10−3, so log10(0.001) = −3.
Common trap: (B) drops the sign. Logs of numbers between 0 and 1 are negative.
LogarithmsIdentitiesln xThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
A complex number a + bi is a point on a plane: a across, b up.
What are the magnitude and angle of −1 + i√3?
r = √(1 + 3) = 2. The point (−1, √3) is in quadrant II, so θ = 180° − 60° = 120°. Answer: 2∠120°.
Common trap: (D) is what arctan(√3/−1) returns on a calculator. Always check the quadrant.
Complex Numberspolar coordinate systemEulerThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
To multiply matrices, take each row of the first times each column of the second.
Let A have rows (1, 2), (3, 4) and B have rows (0, 1), (1, 0). The product AB has rows:
Row 1: (1·0 + 2·1, 1·1 + 2·0) = (2, 1). Row 2: (3·0 + 4·1, 3·1 + 4·0) = (4, 3). AB = (2, 1), (4, 3). Multiplying by B on the right swaps the columns.
Common trap: (D) is BA, which swaps the rows instead. Order matters.
MatricesDeterminantsmatrix multiplicationThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
A system of linear equations can be written Ax = b and solved as x = A−1b.
Solve 2x + y = 7 and x − y = −1.
Add the equations: 3x = 6, so x = 2. Then y = 7 − 4 = 3. x = 2, y = 3.
Common trap: (B) satisfies the second equation only. Always check both.
Matrix inverseCrameradjointThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
The dot product turns two vectors into a number, A·B = |A||B| cos θ, which gives angles, projections and work.
What is A × B for A = (1, 2, 3) and B = (4, 5, 6)?
(aybz − azby, azbx − axbz, axby − aybx) = (12 − 15, 12 − 6, 5 − 8) = (−3, 6, −3).
Common trap: (D) is B × A, and (B) is the dot product.
Vectorsdot productcross productThat was 1 of 4 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
An arithmetic progression adds the same amount d each step; a geometric progression multiplies by the same ratio r.
A geometric progression starts at 2 with ratio 3. What is the 6th term?
l = arn−1 = 2(3)5 = 2(243) = 486.
Common trap: (C) uses rn instead of rn−1.
ProgressionsArithmetic ProgressionGeometric ProgressionThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
Newton’s Notes covers all 14 FE Mechanical exam areas: lessons, worked examples, hundreds of practice problems with full solutions, a 110-question practice exam, calculator keystrokes and Ask Newton, the AI tutor. Try it free for 3 days.
Start your 3-day free trialSee plans →