Model Comparison: Number & Quantity

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Gemini 3 Pro

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What You Need to Know

• Complex Numbers are Routine: You must be comfortable treating the imaginary unit i as a variable during algebra, remembering that i^2 = -1.
• Exponents Rules are Essential: Fluency in converting between radical notation (\sqrt[n]{x}) and rational exponents (x^{1/n}) is required for both simplification and solving equations.
• Matrices & Vectors follow specific rules: These topics are not difficult, but they require memorizing specific algorithms for operations like matrix multiplication and vector addition.
• Quantities: Pay attention to units and scientific notation, as the ACT often tests your ability to manipulate large and small numbers efficiently.

Real and Complex Number Systems

The Real Number System

Every number on the continuous number line is a Real Number. On the ACT, you must distinguish between:

• Rational Numbers: Can be written as a fraction \frac{p}{q} where p and q are integers and q \neq 0. Their decimal expansions either terminate (e.g., 0.75) or repeat (e.g., 0.333…).
• Irrational Numbers: Cannot be written as a simple fraction. Their decimals go on forever without repeating (e.g., \pi, \sqrt{2}, e).

Complex Numbers

A Complex Number takes the form a + bi, where:

• a is the real part.
• b is the imaginary part.
• i = \sqrt{-1}.

Operations with Complex Numbers

Treat i exactly like a variable (like x) for addition, subtraction, and multiplication. The only difference is that you must simplify i^2 to -1 whenever it appears.

• Addition/Subtraction: Combine real parts with real parts, and imaginary parts with imaginary parts.
  (3 + 2i) - (1 - 4i) = (3 - 1) + (2i - (-4i)) = 2 + 6i

• Multiplication: Use FOIL (First, Outer, Inner, Last).
  (2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2
  = 8 + 10i - 3(-1)
  = 8 + 10i + 3 = 11 + 10i

• Powers of i: The powers of i repeat in a cycle of 4:

  • i^1 = i
  • i^2 = -1
  • i^3 = -i
  • i^4 = 1

  To find a high power like i^{25}, divide the exponent by 4 and look at the remainder.
  25 \div 4 = 6 with a remainder of 1. Therefore, i^{25} = i^1 = i.

• Complex Conjugate: The conjugate of a + bi is a - bi. Multiplying a complex number by its conjugate always results in a real number:
  (a + bi)(a - bi) = a^2 + b^2

Exam Focus

• Why it matters: Complex number questions are "easy points" if you know the rules, but impossible if you don't. They usually appear in the first 30 questions.
• Typical question patterns:
  • Simplifying an expression involving i (e.g., "For i = \sqrt{-1}, what is the value of…?").
  • Multiplying two binomials containing i.
  • Rationalizing a denominator (multiplying top and bottom by the conjugate).
• Common mistakes: Forgetting to distribute the negative sign when subtracting complex numbers, or leaving the answer as i^2 instead of -1.

Integer and Rational Exponents

Laws of Exponents

Mastering these rules allows you to simplify expressions quickly.

• Product Rule: x^a \cdot x^b = x^{a+b}
• Quotient Rule: \frac{x^a}{x^b} = x^{a-b}
• Power Rule: (x^a)^b = x^{a \cdot b}
• Zero Exponent: x^0 = 1 (for x \neq 0)
• Negative Exponent: x^{-a} = \frac{1}{x^a} and \frac{1}{x^{-a}} = x^a

Rational (Fractional) Exponents

A fractional exponent represents a root (radical). The denominator of the fraction is the index of the root.

x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m

Examples:

• x^{1/2} = \sqrt{x}
• 8^{2/3} = (\sqrt[3]{8})^2 = (2)^2 = 4

Scientific Notation

Used to express very large or very small numbers. Format: a \times 10^n, where 1 \le |a| < 10.

• Multiplication: (a \times 10^n)(b \times 10^m) = (a \cdot b) \times 10^{n+m}
• Division: \frac{a \times 10^n}{b \times 10^m} = (\frac{a}{b}) \times 10^{n-m}

Exam Focus

• Why it matters: Exponents are fundamental to algebra. The ACT explicitly tests your ability to switch forms (radical to exponent) without a calculator.
• Typical question patterns:
  • "Which of the following is equivalent to…" usually involving converting radicals to fractional exponents.
  • Solving equations where the variable is in the exponent (e.g., 27^{x} = 9^{x+2}). Strategy: express both sides with the same base (3 in this case).
• Common mistakes: Confusing negative exponents with negative numbers (2^{-3} is \frac{1}{8}, not -8). Multiplying bases instead of adding exponents (x^2 \cdot x^3 is x^5, not x^6).

Vectors

Vectors are quantities that have both magnitude (length) and direction. On the ACT, vectors are usually represented algebraically as component vectors \langle a, b \rangle or using the standard unit vectors i and j (where i is horizontal, j is vertical). Note: In vector context, i and j are not imaginary numbers.

Component Form

A vector \mathbf{v} = \langle x, y \rangle starts at the origin (0,0) and ends at point (x,y).

• x is the horizontal component.
• y is the vertical component.
• Alternative notation: \mathbf{v} = x\mathbf{i} + y\mathbf{j}

Vector Operations

• Addition: Add corresponding components.
  \langle x1, y1 \rangle + \langle x2, y2 \rangle = \langle x1+x2, y1+y2 \rangle
  Graphically: The "Head-to-Tail" method. Place the tail of the second vector at the head of the first. The resultant vector goes from the start of the first to the end of the second.

• Subtraction: Subtract corresponding components.
  \langle x1, y1 \rangle - \langle x2, y2 \rangle = \langle x1-x2, y1-y2 \rangle

• Scalar Multiplication: Distribute the scalar (constant number) to both components.
  k \cdot \langle x, y \rangle = \langle kx, ky \rangle
  Graphically: This changes the length of the vector. If k is negative, it reverses the direction.

Exam Focus

• Why it matters: Vectors appear infrequently (maybe 1 question per test), but the math is extremely simple if you know the notation.
• Typical question patterns:
  • Given vector \mathbf{u} and vector \mathbf{v}, find \mathbf{u} + \mathbf{v}.
  • Word problems involving forces or velocity (e.g., a plane flying North with a wind blowing East).
• Common mistakes: Confusing the vector i (horizontal unit vector) with the imaginary number i. The context of the question will make it clear.

Matrices and Matrix Operations

A Matrix is a rectangular array of numbers arranged in rows and columns. Dimensions are given as Rows \times Columns (R \times C).

Matrix Operations

• Addition/Subtraction: You can only add or subtract matrices if they have the exact same dimensions. Simply add or subtract the numbers in the corresponding positions.
  \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}

• Scalar Multiplication: Multiply every single entry in the matrix by the scalar number.
  2 \cdot \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}

• Matrix Multiplication: You can multiply Matrix A and Matrix B only if the columns of A equal the rows of B.

  • If A is m \times n and B is n \times p, the product AB is defined and will have dimensions m \times p.
  • How to multiply: Multiply ROW by COLUMN. For the entry in row 1, column 1 of the answer, multiply the elements of Row 1 of the first matrix by the corresponding elements of Column 1 of the second matrix and add them up.

Determinant

The determinant is a value associated with a square matrix. For a 2 \times 2 matrix:
Det \begin{bmatrix} a & b \ c & d \end{bmatrix} = ad - bc

Exam Focus

• Why it matters: The ACT often includes 1-2 matrix questions. They rely heavily on definitions (like when multiplication is possible).
• Typical question patterns:
  • Calculating the determinant of a 2 \times 2 matrix.
  • "Is the product AB defined? If so, what are its dimensions?"
  • Simple addition or scalar multiplication.
• Common mistakes: Thinking matrix multiplication is commutative. AB is almost never equal to BA. In fact, BA might not even be defined when AB is.

Quick Review Checklist

• Can you simplify i^{35} or i^{102} using the cycle of 4?
• Do you know that i^2 = -1 and how to use it when multiplying complex numbers?
• Can you convert \sqrt[3]{x^5} into x^{5/3} instantly?
• Can you calculate the result of 3\mathbf{u} - 2\mathbf{v} given vectors \mathbf{u} and \mathbf{v}?
• Do you know the formula for the determinant of a 2 \times 2 matrix (ad - bc)?
• Can you determine the dimensions of a product matrix (e.g., 3 \times 2 multiplied by 2 \times 4 equals 3 \times 4)?

Final Exam Pitfalls

1. The Undefined Matrix Product Trap

   • Mistake: Assuming you can multiply any two matrices.
   • Correction: Always check dimensions first. Inner numbers must match (2\times\mathbf{3} and \mathbf{3}\times4 works; 2\times\mathbf{3} and \mathbf{2}\times3 does not).

2. The Negative Exponent Fraction Flip

   • Mistake: Thinking x^{-2} means a negative number like -x^2.
   • Correction: Negative exponents create reciprocals. 5^{-2} = \frac{1}{5^2} = \frac{1}{25}. The value is positive and small.

3. The Complex Number Sign Error

   • Mistake: When calculating the determinant ad - bc, forgetting that subtracting a negative is addition.
   • Correction: If ad = 10 and bc = -4, the determinant is 10 - (-4) = 14, not 6.

4. Vector Direction Signs

   • Mistake: Ignoring signs when calculating vector components (\langle -3, 4 \rangle vs \langle 3, 4 \rangle).
   • Correction: Sketch the vector roughly on a coordinate plane to ensure your resulting direction makes sense (e.g., left and up for negative x, positive y).

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GPT 5.2 Pro

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What You Need to Know

• You must be fluent moving between number forms—fractions, decimals, radicals, and complex numbers—and know when each form is most useful.
• Exponent rules (including rational exponents) are a frequent source of quick points—if you simplify cleanly and respect domain restrictions.
• Vectors and matrices are less common on ACT than core algebra/geometry, but when they appear they’re usually procedural: compute components, multiply matrices, or apply a simple transformation.
• On a multiple-choice exam, strategic methods (plugging in, checking answer choices, and estimation) often beat long algebra—especially with radicals, exponents, and matrices.

Real and Complex Number Systems

What they are

• Real numbers are all numbers on the number line—this includes rational numbers and irrational numbers.
  • Rational: can be written as a ratio \frac{p}{q} where p,q are integers and q\neq 0.
  • Irrational: cannot be expressed as such a fraction (e.g., many square roots of non-perfect squares).
• Complex numbers extend the real numbers using the imaginary unit i defined by:
  i^2=-1
  A complex number has the form:
  a+bi
  where a is the real part and b is the imaginary part.

Key skills and properties

Classifying numbers (common ACT task)

• Integers: \ldots,-2,-1,0,1,2,\ldots
• Rational examples: \frac{7}{8}, -3, 0.125, 0.\overline{3}
• Irrational examples: \sqrt{2}, \pi

Complex arithmetic

• Add/subtract like terms:
  (a+bi)+(c+di)=(a+c)+(b+d)i
• Multiply (FOIL):
  (a+bi)(c+di)=ac+adi+bci+bdi^2=(ac-bd)+(ad+bc)i
• Complex conjugate of a+bi is a-bi.
  • Useful for division:
    \frac{a+bi}{c+di}\cdot\frac{c-di}{c-di}=\frac{(a+bi)(c-di)}{c^2+d^2}

Square roots of negatives

• For k\ge 0:
  \sqrt{-k}=i\sqrt{k}

Worked examples

Example 1 (classify): Is 0.\overline{27} rational?

• A repeating decimal is rational because it can be written as a fraction.
• So 0.\overline{27} is rational (and therefore real).

Example 2 (divide complex numbers): Simplify \frac{3+4i}{1-2i}.

• Multiply by the conjugate 1+2i:
  \frac{3+4i}{1-2i}\cdot\frac{1+2i}{1+2i}=\frac{(3+4i)(1+2i)}{(1-2i)(1+2i)}
• Denominator:
  (1-2i)(1+2i)=1-(2i)^2=1-(-4)=5
• Numerator:
  (3+4i)(1+2i)=3+6i+4i+8i^2=3+10i-8=-5+10i
• Result:
  \frac{-5+10i}{5}=-1+2i

Exam Focus

• Why it matters: ACT questions often mix number types with algebraic manipulation—errors with i or rational/irrational properties quickly derail otherwise-easy problems.
• Typical question patterns:
  • Simplify expressions involving \sqrt{-k} or add/multiply complex numbers.
  • Identify whether a number is rational/irrational (often via decimal behavior or radicals).
  • Rationalize or simplify results to match answer choices.
• Common mistakes:
  • Treating \sqrt{a+b} as \sqrt{a}+\sqrt{b} (not valid in general).
  • Forgetting i^2=-1 (especially in multiplication).
  • Dividing complex numbers without multiplying by the conjugate.

Integer and Rational Exponents

Core idea

• Integer exponents represent repeated multiplication:
  a^n=\underbrace{a\cdot a\cdot\ldots\cdot a}_{n\text{ times}}
• Zero and negative exponents:
  a^0=1\quad (a\neq 0)
  a^{-n}=\frac{1}{a^n}
• Rational exponents connect exponents and radicals:
  a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m
  (For ACT-style real-number answers, you typically assume principal real roots where defined.)

Exponent rules (high-frequency)

For a\neq 0 and appropriate exponents:

• Product: a^m\cdot a^n=a^{m+n}
• Quotient: \frac{a^m}{a^n}=a^{m-n}
• Power of a power: (a^m)^n=a^{mn}
• Power of a product: (ab)^n=a^n b^n
• Power of a quotient: \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}

Domain + sign cautions (especially with even roots)

• \sqrt[n]{a} is real for even n only if a\ge 0.
• A common trap is confusing:
  \sqrt{x^2}=|x|
  not necessarily x.

Worked examples

Example 1 (simplify): Simplify \left(16\right)^{\frac{3}{4}}.

• Compute the fourth root first: 16^{\frac{1}{4}}=2 because 2^4=16.
• Then cube: 16^{\frac{3}{4}}=\left(16^{\frac{1}{4}}\right)^3=2^3=8.

Example 2 (solve with exponents): Solve 2^{x+1}=16.

• Rewrite 16 as a power of 2: 16=2^4.
• Equate exponents: x+1=4 so x=3.

Exam Focus

• Why it matters: Exponent manipulation is a staple tool inside algebra, functions, and even geometry formulas—ACT often tests it indirectly.
• Typical question patterns:
  • Simplify expressions with negative/rational exponents to match a clean answer choice.
  • Solve equations by rewriting both sides with the same base.
  • Convert between radicals and rational exponents.
• Common mistakes:
  • Adding exponents when bases differ (e.g., 2^a\cdot 3^a\neq 6^{2a}).
  • Dropping absolute value in \sqrt{x^2}=|x|.
  • Mishandling negative exponents—forgetting they create reciprocals.

Vectors

What they are (ACT-relevant view)

• A vector represents a quantity with magnitude and direction.
• In coordinate form, a 2D vector is often written as:
  \langle a,b\rangle
  or as a directed segment from point A(x1,y1) to B(x2,y2):
  \overrightarrow{AB}=\langle x2-x1,\;y2-y1\rangle

Basic operations

• Addition:
  \langle a,b\rangle+\langle c,d\rangle=\langle a+c,\;b+d\rangle
• Scalar multiplication:
  k\langle a,b\rangle=\langle ka,\;kb\rangle
• Magnitude (length):
  |\langle a,b\rangle|=\sqrt{a^2+b^2}

Useful connections to geometry/physics

• Displacement: moving from A to B uses \overrightarrow{AB}.
• Resultant motion: adding vectors models combined movements (e.g., wind + walking direction).

Worked examples

Example 1 (vector from two points): Find \overrightarrow{AB} for A(2,-1) and B(-3,5).

• Compute components:
  \overrightarrow{AB}=\langle -3-2,\;5-(-1)\rangle=\langle -5,\;6\rangle

Example 2 (magnitude): Find |\langle 6,-8\rangle|.

• Use the magnitude formula:
  \sqrt{6^2+(-8)^2}=\sqrt{36+64}=\sqrt{100}=10

Exam Focus

• Why it matters: When vectors appear on ACT, they’re usually a fast coordinate-geometry check—components and magnitude tie directly to slope/distance ideas.
• Typical question patterns:
  • Compute a vector between two points or add/subtract vectors.
  • Find magnitude and match it to a distance.
  • Use vector ideas implicitly through displacement problems.
• Common mistakes:
  • Reversing direction: \overrightarrow{AB}=-\overrightarrow{BA}.
  • Mixing up adding points vs adding vectors (points are locations; vectors are differences).
  • Forgetting to square both components in magnitude.

Matrices and Matrix Operations

What they are

• A matrix is a rectangular array of numbers. A typical entry is written as a_{ij} (row i, column j).
• Dimensions matter: a matrix with m rows and n columns is m\times n.

Core operations

Addition/subtraction (same dimensions only)

If A and B are both m\times n:
A\pm B is found by adding/subtracting corresponding entries.

Scalar multiplication

kA multiplies every entry of A by k.

Matrix multiplication (order matters)

If A is m\times n and B is n\times p, then AB is m\times p.

• Entry rule:
  (AB){ij}=\sum{k=1}^{n} a{ik}b{kj}
• In general:
  AB\neq BA

Determinant of a 2\times 2 (common when tested)

For \begin{pmatrix}a & b\ c & d\end{pmatrix}:
\det=ad-bc

Inverse of a 2\times 2 (only if determinant nonzero)

If ad-bc\neq 0, then:
\begin{pmatrix}a & b\ c & d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d & -b\ -c & a\end{pmatrix}

Real-world/application lens

• Matrices can represent systems (input-output rules) and transformations (like stretches/rotations). On ACT, this is usually computational: apply a given matrix rule to a vector.

Worked examples

Example 1 (multiply): Compute
\begin{pmatrix}1 & 2\ 3 & 4\end{pmatrix}\begin{pmatrix}5\ 6\end{pmatrix}

• Multiply rows by the column:
  \begin{pmatrix}1\cdot 5+2\cdot 6\ 3\cdot 5+4\cdot 6\end{pmatrix}=\begin{pmatrix}17\ 39\end{pmatrix}

Example 2 (determinant): Find the determinant of
\begin{pmatrix}-2 & 7\ 1 & 3\end{pmatrix}

• Compute:
  (-2)(3)- (7)(1)=-6-7=-13

Exam Focus

• Why it matters: Matrix questions (when present) reward careful bookkeeping—dimensions, order, and arithmetic accuracy.
• Typical question patterns:
  • Multiply a 2\times 2 matrix by a vector (column) or another 2\times 2 matrix.
  • Compute a 2\times 2 determinant.
  • Apply a matrix “rule” to transform coordinates.
• Common mistakes:
  • Attempting multiplication with incompatible dimensions.
  • Treating matrix multiplication as entrywise multiplication.
  • Swapping order and assuming AB=BA.

Quick Review Checklist

• Can you classify numbers as integer, rational, irrational, real, or complex?
• Can you simplify expressions involving i and use conjugates to divide complex numbers?
• Can you apply exponent rules, including negative and rational exponents, without breaking domain restrictions?
• Can you convert between radicals and rational exponents (e.g., a^{\frac{m}{n}} forms)?
• Can you compute \overrightarrow{AB} from two points and find a vector’s magnitude?
• Can you add vectors and scale vectors correctly?
• Can you add matrices and multiply a matrix by a scalar?
• Do you know when matrix multiplication AB is defined, and can you compute it for small matrices?
• Can you compute the determinant of a 2\times 2 matrix?

Final Exam Pitfalls

1. Forgetting rules for negatives and roots: Use \sqrt{-k}=i\sqrt{k} and remember \sqrt{x^2}=|x| when variables can be negative.
2. Misusing exponent rules across different bases: Only combine exponents directly when bases match (e.g., a^m\cdot a^n=a^{m+n}).
3. Dropping parentheses with exponents: Distinguish (-3)^2=9 from -3^2=-9.
4. Reversing vector direction: Compute \overrightarrow{AB}=\langle x2-x1,\;y2-y1\rangle in the stated order.
5. Matrix dimension/order errors: Check sizes before multiplying and don’t assume AB=BA—write dimensions and compute systematically.

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Claude Opus 4.6

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What You Need to Know

• The ACT Math section tests Number & Quantity concepts across roughly 10–15% of the 60-question test. These questions range from basic real number properties to more advanced topics like complex numbers, vectors, and matrices — especially on the harder end of the test.
• Exponent rules are among the most frequently tested skills in this category. You must be fluent with integer, rational, and negative exponents to avoid costly errors under time pressure.
• Complex numbers, vectors, and matrices appear primarily in the upper difficulty range (questions 45–60). Knowing even the basics here can earn you points that many students leave on the table.
• Most Number & Quantity questions reward quick, accurate arithmetic and rule application rather than deep conceptual reasoning. Memorize the rules, practice applying them, and you'll gain speed.

Real and Complex Number Systems

The Real Number System

The real numbers include all numbers on the number line. You should know the hierarchy:

Key properties to remember:

• Every integer is rational (e.g., 5 = \frac{5}{1}).
• \sqrt{n} is irrational when n is not a perfect square.
• \pi and e are irrational.

Complex Numbers

A complex number has the form:

a + bi

where a is the real part, b is the imaginary part, and i = \sqrt{-1}.

Essential powers of i — these cycle every four:

To evaluate i^n for large n, divide n by 4 and use the remainder. For example, i^{27}: 27 \div 4 = 6 remainder 3, so i^{27} = i^3 = -i.

Operations with complex numbers:

• Addition/Subtraction: Combine like parts. (3 + 2i) + (1 - 5i) = 4 - 3i
• Multiplication: Use FOIL and replace i^2 with -1.
  (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i^2 = 2 + i - 3(-1) = 5 + i
• Complex conjugate of a + bi is a - bi. Multiply by the conjugate to rationalize denominators.

Exam Focus

• Why it matters: Complex number questions appear 1–3 times per ACT, often in the second half. They are quick points if you know the rules.
• Typical question patterns:
  • Simplify a power of i (e.g., i^{53})
  • Multiply or divide two complex numbers
  • Identify the real or imaginary part of an expression
• Common mistakes:
  • Forgetting that i^2 = -1 during multiplication — always substitute immediately
  • Confusing \sqrt{-9} as -3 instead of 3i
  • Misidentifying 0 as "not rational" — it is rational

Integer and Rational Exponents

Core Exponent Rules

You must have these rules memorized cold:

Rational (Fractional) Exponents

Rational exponents connect exponents to radicals:

a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m

Example: Simplify 8^{\frac{2}{3}}.

Step 1: Rewrite as \left(\sqrt[3]{8}\right)^2.

Step 2: \sqrt[3]{8} = 2.

Step 3: 2^2 = 4.

So 8^{\frac{2}{3}} = 4.

Memory aid: In a^{\frac{m}{n}}, think "power over root" — the numerator is the power, the denominator is the root.

Exam Focus

• Why it matters: Exponent questions are among the most common on the ACT Math section. They appear at every difficulty level.
• Typical question patterns:
  • Simplify an expression with multiple exponent rules in one problem
  • Convert between radical and exponent notation
  • Solve equations like 2^{x} = 16
• Common mistakes:
  • Adding exponents when bases are different (2^3 \cdot 3^2 \neq 6^5)
  • Treating a^0 as 0 instead of 1
  • Flipping the meaning of numerator and denominator in rational exponents

Vectors

What Is a Vector?

A vector is a quantity with both magnitude (length) and direction. On the ACT, vectors are typically represented in component form:

\vec{v} = \langle a, b \rangle

where a is the horizontal component and b is the vertical component.

Key Operations

• Addition: \langle a, b \rangle + \langle c, d \rangle = \langle a + c,\; b + d \rangle
• Subtraction: \langle a, b \rangle - \langle c, d \rangle = \langle a - c,\; b - d \rangle
• Scalar multiplication: k \langle a, b \rangle = \langle ka,\; kb \rangle
• Magnitude: |\vec{v}| = \sqrt{a^2 + b^2}

Example: If \vec{u} = \langle 3, -1 \rangle and \vec{w} = \langle -2, 4 \rangle, find \vec{u} + \vec{w}.

\vec{u} + \vec{w} = \langle 3 + (-2),\; -1 + 4 \rangle = \langle 1, 3 \rangle

Direction and Real-World Context

Vectors model forces, velocity, and displacement. An ACT question might describe a boat moving at a certain speed and angle and ask you to find its horizontal or vertical component using:

vx = |\vec{v}| \cos\theta, \quad vy = |\vec{v}| \sin\theta

Exam Focus

• Why it matters: Vector questions are relatively rare on the ACT (0–2 per test) but tend to be straightforward if you know component operations.
• Typical question patterns:
  • Add or subtract two vectors in component form
  • Find the magnitude of a resultant vector
  • Interpret a word problem involving direction and speed
• Common mistakes:
  • Subtracting components in the wrong order
  • Forgetting to square both components before adding under the square root for magnitude
  • Confusing vector notation \langle a, b \rangle with coordinate point notation (a, b)

Matrices and Matrix Operations

What Is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with m rows and n columns is called an m \times n matrix.

A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}

This is a 2 \times 2 matrix.

Key Operations

Addition/Subtraction — matrices must have the same dimensions. Add or subtract corresponding entries:

\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}

Scalar Multiplication — multiply every entry by the scalar:

3 \begin{bmatrix} 1 & 2 \ 0 & -1 \end{bmatrix} = \begin{bmatrix} 3 & 6 \ 0 & -3 \end{bmatrix}

Matrix Multiplication — for A{m \times n} and B{n \times p}, the product AB is m \times p. The entry in row i, column j is the dot product of row i of A and column j of B.

Example:

\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 \ 6 \end{bmatrix} = \begin{bmatrix} 1(5) + 2(6) \ 3(5) + 4(6) \end{bmatrix} = \begin{bmatrix} 17 \ 39 \end{bmatrix}

Critical rule: Matrix multiplication is not commutative — AB \neq BA in general.

Determinant of a 2×2 Matrix

For A = \begin{bmatrix} a & b \ c & d \end{bmatrix}:

\det(A) = ad - bc

This occasionally appears on the ACT and is a quick calculation.

Exam Focus

• Why it matters: Matrix questions appear 0–2 times per ACT, usually at the medium-to-hard level. They are mechanical and very scoreable.
• Typical question patterns:
  • Add two matrices or perform scalar multiplication
  • Multiply a 2 \times 2 matrix by a 2 \times 1 matrix
  • Find a specific entry of a product matrix
  • Calculate a 2 \times 2 determinant
• Common mistakes:
  • Trying to add matrices of different dimensions
  • Multiplying corresponding entries instead of using dot products for matrix multiplication
  • Assuming AB = BA

Quick Review Checklist

• Can you classify a number as natural, whole, integer, rational, irrational, or real?
• Can you simplify any power of i using the cycle of four?
• Can you multiply and divide complex numbers, including using the conjugate?
• Can you apply all six core exponent rules quickly and correctly?
• Can you convert between radical notation and rational exponent notation?
• Can you add, subtract, and scalar-multiply vectors in component form?
• Can you calculate the magnitude of a vector?
• Can you add and subtract matrices of equal dimensions?
• Can you multiply two matrices and determine the dimensions of the product?
• Do you know the formula for the determinant of a 2 \times 2 matrix?

Final Exam Pitfalls

1. Mishandling i^2 in complex multiplication. When you FOIL complex numbers, you will always get an i^2 term. Replace it with -1 immediately — don't leave it or treat it as positive 1.

2. Applying the product exponent rule to different bases. 2^3 \cdot 5^3 = (2 \cdot 5)^3 = 10^3 is valid, but 2^3 \cdot 3^2 cannot be simplified by adding exponents. Exponent rules require the same base (or the same exponent for the product-to-a-power rule).

3. Treating a^0 as 0. Any nonzero base raised to the zero power equals 1, not 0.

4. Multiplying matrix entries element-by-element. Matrix multiplication uses dot products of rows and columns. Element-wise multiplication is a completely different operation that the ACT does not use.

5. Forgetting dimension requirements. You can only add/subtract matrices with identical dimensions. You can only multiply A{m \times n} \cdot B{n \times p} when the inner dimensions match (n = n). If a question asks whether an operation is possible, check dimensions first.

6. Computing \sqrt{a^2 + b^2} as a + b for vector magnitude. The square root of a sum is not the sum of the square roots. Always compute a^2 + b^2 first, then take the square root of the total.