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Vectors & Matrices HL

IB Mathematics: Applications & Interpretation · Topic 3: Geometry & Trigonometry / Topic 1: Algebra
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Vector Operations
A vector has magnitude and direction. Written as a (bold) or \( a⃗ \) (arrow notation). Components in 2D: (\( a_1, a_2) \) or 3D: (\( a_1, a_2, a_3) \).
a b a + b Triangle rule for vector addition
Addition / Subtraction
\( a + b = (a_1+b_1, a_2+b_2, a_3+b_3) \)
Add/subtract corresponding components
Scalar multiplication
\( ka \) = (\( ka_1, ka_2, ka_3) \)
Scales magnitude by |\( k|, \) reverses direction if \( k \) < 0
Magnitude
|a| = \( \sqrt{a_1^2 + a_2^2 + a_3^2} \)
Unit vector
\( â = a \) / |a|
Magnitude = 1, same direction
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Dot Product (Scalar Product)
Component form
a \( \cdot \) b = \( a_1b_1 + a_2b_2 + a_3b_3 \)
Geometric form
a \( \cdot \) b = |a||b| cos \( \theta \)
Finding the angle between two vectors
\( \cos \theta = (a \) \( \cdot \) b) / (|a||b|)
If a \( \cdot \) b = 0, the vectors are perpendicular
a b θ cos θ = (a·b) / (|a||b|)
Worked Example
Find the angle between a = (2, 1, \( -3) \) and b = (4, \( -2, 1) \).
a \( \cdot \) b = 2(4) + 1(\( -2) + (-3)(1) = 8 - 2 - 3 = 3 \)
|a| = \( \sqrt{4+1+9} = \sqrt{14}, \)   |b| = \( \sqrt{16+4+1} = \sqrt{21} \)
cos \( \theta = 3 / (\sqrt{14} \times \sqrt{21}) = 3/\sqrt{294} = 0.17496... \)
Answer: \( \theta = 79.9 \)° (3 s.f.)
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Cross Product (Vector Product)
Cross product (3D only)
a \( \times \) b = (\( a_2b_3 - a_3b_2, \)   \( a_3b_1 - a_1b_3, \)   \( a_1b_2 - a_2b_1) \)
Magnitude of cross product
|a \( \times \) b| = |a||b| sin \( \theta \)
Area of parallelogram
Area = |a \( \times \) b|
Triangle area = \( \frac{1}{2}| \)a \( \times \) b|
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Direction: a \( \times \) b is perpendicular to both a and b (right-hand rule). Also a \( \times \) b = \( -( \)b \( \times \) a).
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Vector Equation of a Line
Line through point a in direction b
\( r = a \) + \( \lambda \)b
a = position vector of a point on the line, b = direction vector, \( \lambda \in \mathbb{R} \)
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Parallel lines: direction vectors are scalar multiples. Intersecting lines: solve for \( \lambda \) and \( \mu \) — if all components are consistent, they intersect. Skew lines: not parallel and do not intersect (3D only).
JMaths · www.jmaths.xyz Vectors & Matrices · Page 1 of 2

Vectors & Matrices HL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra
www.jmaths.xyz
5
Matrix Operations
Matrices are rectangular arrays of numbers. Multiplication is not commutative: AB \( \neq \) BA in general.
Determinant (2\( \times 2) \)
\( det(A) = |A| = ad - bc \)
For A = [\( a b \) ; \( c d] \)
Inverse (2\( \times 2) \)
\( A^{-1} = (1/det(A)) [d -b ; -c a] \)
Exists only if det(A) \( \neq 0 \)
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Solving systems: Ax = b \( \to \) x = A\( -1 \)b. Use the GDC for 3\( \times 3 \) systems — finding inverses by hand is not expected.
GDC: Matrices
TI-84 Plus CE
[2ND][MATRIX] → Edit → enter matrix dimensions and values
Multiply: [A] \( \times \) [B]    Inverse: [A][x\( -1 \)]
Determinant: [2ND][MATRIX] → MATH → det([A])
TI-Nspire CX II
Define: a := [[2,3][1,4]] (use template or type)
Inverse: \( a^{-1} \)    Determinant: det(a)
Solve system: simultEqn() or use rref()
Casio fx-CG50
[MENU] → Run-Matrix → [F3] (MAT) to enter matrices
Multiply: Mat A \( \times \) Mat B    Inverse: Mat A\( -1 \)
Determinant: det(Mat A)
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Eigenvalues & Eigenvectors (2\( \times 2) \)
If Av = \( \lambda \)v, then \( \lambda \) is an eigenvalue and v is the corresponding eigenvector.
Finding eigenvalues
\( det(A - \lambda I) = 0 \)
For 2\( \times 2: (\( a - \lambda )(d - \lambda ) - bc = 0 \to \) solve the quadratic for \( \lambda \)
Steps: (1) Solve det(A \( - \lambda I) = 0 \) for eigenvalues. (2) For each \( \lambda , \) solve (A \( - \lambda I) \)v = 0 for the eigenvector.
Worked Example
Find eigenvalues of A = [3 1 ; 0 2].
det(A \( - \lambda I) = (3-\lambda )(2-\lambda ) - 0 = 0 \)
\( \lambda ^2 - 5\lambda + 6 = 0 \to (\lambda - 2)(\lambda - 3) = 0 \)
Answer: \( \lambda = 2 \) and \( \lambda = 3 \)
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Matrix Transformations
A 2\( \times 2 \) matrix transforms points in the plane. Apply by multiplying: image = M \( \times \) point.
Rotation by \( \theta \) anticlockwise
\( [\cos\theta -\sin\theta ; \sin\theta \cos\theta ] \)
Reflection in \( y = x \)
[0 1 ; 1 0]
Reflection in \( x- \)axis
\( [1 0 ; 0 -1] \)
Enlargement, scale factor \( k \)
\( [k 0 ; 0 k] \)
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Determinant meaning: |det(M)| gives the \( area scale factor \) of the transformation. If det(M) < 0, the transformation reverses orientation.
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Common error: For combined transformations, the order matters. “Reflect then rotate” means multiply R \( \times \) M \( \times \) point (rightmost matrix acts first).
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Common error: Confusing the dot product (scalar result) with matrix multiplication (matrix result). a \( \cdot \) b gives a number; AB gives a matrix.
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Formula booklet: The dot product, cross product, magnitude, and vector line equation are given. Eigenvalue method and transformation matrices are NOT in the booklet — learn the process.
JMaths · www.jmaths.xyz Vectors & Matrices · Page 2 of 2