β Click an answer β correct: π confetti! | wrong: π explanation shown
β Each question has a memory keyword to help you explain it in English!
π Table of Contents
β Vectors (Q1βQ5)Magnitude Β· Direction Β· Dot Product
MIND Β· Magnitude | Inverse (negative) | Not scalar | Direction matters β A vector has BOTH size AND direction. A scalar has only size.
π Quick Example
If \(\vec{a} = \langle 3, 4 \rangle\), then \(|\vec{a}| = \sqrt{3^2+4^2} = \sqrt{25} = 5\) β Use the Pythagorean theorem! The magnitude IS the hypotenuse.
1
π― VECTORS Β· Magnitude
β Easy β but students write the formula wrong!
Find the magnitude of vector \(\vec{v} = \langle -5,\; 12 \rangle\). β οΈ Tricky: negative components don't make magnitude negative!
2
π― VECTORS Β· Direction Angle
β Easy β but many students forget which quadrant!
Vector \(\vec{u} = \langle -4,\; 4 \rangle\). What is the direction angle \(\theta\) (measured from positive x-axis)? β οΈ \(\tan^{-1}\) alone won't give the right quadrant β check signs!
Given \(\vec{a} = \langle 2,\; -1 \rangle\) and \(\vec{b} = \langle -3,\; 4 \rangle\), calculate \(2\vec{a} - 3\vec{b}\).
5
π― VECTORS Β· Unit Vector
β β Medium β formula correct but arithmetic slips!
Find the unit vector in the direction of \(\vec{w} = \langle 6,\; 8 \rangle\). π‘ UNIT = divide every component by the magnitude. Result always has magnitude 1.
π Section 2 Β· Matrices
RIDE Β· Row times column | Inverse: flip & divide | Determinant: adβbc | Equal dimensions to add β Matrix multiplication: (mΓn)(nΓp) = (mΓp) β middle must match!
π For 2Γ2: \(\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc\)
\(A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\)
6
π MATRICES Β· Multiplication
β β Medium β Row Γ Column, not element-wise!
Compute \(AB\) where \(A = \begin{pmatrix}1&2\\3&4\end{pmatrix}\) and \(B = \begin{pmatrix}5&0\\1&-1\end{pmatrix}\).
7
π MATRICES Β· Determinant
β Easy β adβbc, but students switch b and c!
Find \(\det(M)\) for \(M = \begin{pmatrix}4&-3\\2&5\end{pmatrix}\). β οΈ det = 0 means the matrix has NO inverse!
8
π MATRICES Β· Inverse Matrix
β β Medium β swap a&d, negate b&c, then divide by det
Find \(A^{-1}\) for \(A = \begin{pmatrix}3&1\\5&2\end{pmatrix}\).
9
π MATRICES Β· Solving System with Matrices
β β Medium β set up matrix equation first!
Use the matrix equation \(AX = B\) to solve:
\(\begin{cases} 2x + y = 7 \\ x + 3y = 11 \end{cases}\)
π‘ \(X = A^{-1}B\) Β· First find \(\det(A)\)!
π Section 3 Β· Polar Graphs
CART Β· Convert with cos/sin | Angle is ΞΈ from x-axis | Radius is r (can be negative!) | Two names for one point β \(x = r\cos\theta,\quad y = r\sin\theta,\quad r^2 = x^2+y^2\)
Convert polar point \(\left(4,\; \dfrac{\pi}{3}\right)\) to rectangular \((x, y)\).
11
π POLAR Β· Rectangular β Polar
β β Medium β always check the quadrant for ΞΈ!
Convert \((-3,\; 3)\) to polar form \((r,\theta)\) with \(r > 0\) and \(0 \le \theta < 2\pi\). β οΈ x is negative, y is positive β Quadrant II!
12
π POLAR Β· Graph Identification
β β Medium β tricky shape names!
The polar equation \(r = 3 + 3\cos\theta\) traces which shape? π‘ If \(a = b\) in \(r = a + b\cos\theta\), the curve passes through the pole (origin).
13
π POLAR Β· Negative r Value
β β β Hard concept β many students skip this!
The polar point \(\left(-2,\; \dfrac{\pi}{6}\right)\) is identical to which of the following? β οΈ Negative r means: go in the opposite direction. Add Ο to the angle!
γ° Section 4 Β· Trigonometry
ASTC Β· All (Q1) | Sin (Q2) | Tan (Q3) | Cos (Q4) β which trig ratio is positive per quadrant Mnemonic: "All Students Take Calculus" π
β Easy β but students confuse sin and cos positions!
Evaluate exactly: \(\sin\!\left(\dfrac{5\pi}{6}\right)\) π‘ Reference angle of \(\tfrac{5\pi}{6}\) is \(\tfrac{\pi}{6}\). Quadrant II β sin is positive!
15
γ° TRIG Β· Pythagorean Identity
β β Medium β use identity, don't find the angle!
If \(\cos\theta = -\dfrac{3}{5}\) and \(\theta\) is in Quadrant III, find \(\sin\theta\). β οΈ In Q3: sin is also negative!
16
γ° TRIG Β· Law of Sines
β β Medium β set up the ratio correctly!
In triangle \(ABC\), \(A = 30Β°\), \(B = 45Β°\), and \(a = 6\). Find side \(b\).
\[\frac{a}{\sin A} = \frac{b}{\sin B}\]
17
γ° TRIG Β· Double Angle Identity
β β Medium β most common identity test question!
If \(\sin\theta = \dfrac{3}{5}\) and \(\theta\) is in Q1, find \(\sin(2\theta)\). π‘ \(\sin(2\theta) = 2\sin\theta\cos\theta\) Β· First find \(\cos\theta\) using Pythagorean identity!
18
γ° TRIG Β· Law of Cosines
β β Medium β use when you have SAS or SSS!
In triangle \(ABC\): \(a=7,\; b=5,\; C=60Β°\). Find side \(c\).
\[c^2 = a^2 + b^2 - 2ab\cos C\]
π₯ Section 5 Β· Mixed Challenge
π These questions connect multiple topics! Think carefully before picking.
19
π₯ MIXED Β· Vector + Trig Connection
β β Medium β vector components use trig!
A force vector has magnitude \(10\) N at angle \(120Β°\) from the positive x-axis.
Express it as a component vector \(\langle F_x,\; F_y \rangle\). π‘ \(F_x = |\vec{F}|\cos\theta,\quad F_y = |\vec{F}|\sin\theta\) β this IS the vector-trig bridge!
20
π₯ MIXED Β· Polar + Trig + Complex Numbers
β β β Hard β Boss level!
Write the complex number \(z = -1 + i\sqrt{3}\) in polar form \(r(\cos\theta + i\sin\theta)\). β οΈ This combines: rectangularβpolar conversion AND trig values. Think quadrant II!