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IB Mathematics Analysis and Approaches Topic AHL… | RevisionDojo

IB
AHL 3.12—Vector definitions

IB Mathematics Analysis and Approaches AHL 3.12—vector Definitions Questionbank

SL 1.1—Using standard form

SL 1.2—Arithmetic sequences and series

SL 1.3—Geometric sequences and series

SL 1.4—Financial apps – compound interest, annual depreciation

SL 1.5—Intro to logs

SL 1.6—Simple proof

SL 1.7—Laws of exponents and logs

SL 1.8—Sum of infinite geo sequence

SL 1.9—Binomial theorem where n is an integer

AHL 1.10—Perms and combs, binomial with negative and fractional indices

AHL 1.11—Partial fractions

AHL 1.12—Complex numbers – Cartesian form and Argand diag

AHL 1.13—Polar and Euler form

AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers

AHL 1.15—Proof by induction, contradiction, counterexamples

AHL 1.16—Solution of systems of linear equations

SL 2.1—Equations of straight lines, parallel and perpendicular

SL 2.2—Functions, notation domain, range and inverse as reflection

SL 2.3—Graphing

SL 2.4—Key features of graphs, intersections using technology

SL 2.5—Composite functions, identity, finding inverse

SL 2.6—Quadratic function

SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots

SL 2.8—Reciprocal and simple rational functions, equations of asymptotes

SL 2.9—Exponential and logarithmic functions

SL 2.10—Solving equations graphically and analytically

SL 2.11—Transformation of functions

AHL 2.12—Factor and remainder theorems, sum and product of roots

AHL 2.13—Rational functions

AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction

AHL 2.15—Solutions of inequalities

AHL 2.16—Graphing modulus equations and inequalities

SL 3.1—3d space, volume, angles, distance, midpoints

SL 3.2—2d and 3d trig, sine rule, cosine rule, area

SL 3.3—Angles of elevation and depression, bearings

SL 3.4—Circle, radians, arcs, sectors

SL 3.5—Unit circle definitions of sin, cos, tan. Exact trig ratios, ambiguous case of sine rule

SL 3.6—Pythagorean identity, double angles

SL 3.7—Circular functions, graphs, composites, transformations

SL 3.8—Solving trig equations

AHL 3.9—Reciprocal trig ratios and their pythagorean identities. Inverse circular functions

AHL 3.10—Compound angle identities

AHL 3.11—Relationships between trig functions

AHL 3.12—Vector definitions

AHL 3.13—Scalar (dot) product

AHL 3.14—Vector equation of line

AHL 3.15—Classification of lines

AHL 3.16—Vector product

AHL 3.17—Vector equations of a plane

AHL 3.18—Intersections of lines & planes

SL 4.1—Concepts, reliability and sampling techniques

SL 4.2—Histograms, CF graphs, box plots

SL 4.3—Mean, median, mode. Mean of grouped data, standard deviation. Quartiles, IQR

SL 4.4—Pearsons, scatter diagrams, eqn of y on x

SL 4.5—Probability concepts, expected numbers

SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams

SL 4.7—Discrete random variables

SL 4.8—Binomial distribution

SL 4.9—Normal distribution and calculations

SL 4.10—X on y regression line

SL 4.11—Conditional and independent probabilities, test for independence

SL 4.12—Z values, inverse normal to find mean and standard deviation

AHL 4.13—Bayes theorem

AHL 4.14—Properties of discrete and continuous random variables

SL 5.1—Introduction of differential calculus

SL 5.2—Increasing and decreasing functions

SL 5.3—Differentiating polynomials, n E Z

SL 5.4—Tangents and normal

SL 5.5—Integration introduction, areas between curve and x axis

SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules

SL 5.7—The second derivative

SL 5.8—Testing for max and min, optimisation. Points of inflexion

SL 5.9—Kinematics problems

SL 5.10—Indefinite integration, reverse chain, by substitution

SL 5.11—Definite integrals, areas under curve onto x-axis and areas between curves

AHL 5.12—First principles, higher derivatives

AHL 5.13—Limits and L’Hopitals

AHL 5.14—Implicit functions, related rates, optimisation

AHL 5.15—Further derivatives and indefinite integration of these, partial fractions

AHL 5.16—Integration by substitution, parts and repeated parts

AHL 5.17—Areas under curve onto y-axis, volume of revolution (about x and y axes)

AHL 5.18—1st order DE’s – Euler method, variables separable, integrating factor, homogeneous DE using sub y=vx

AHL 5.19—Maclaurin series

IB Mathematics Analysis and Approaches (AA) Topic AHL 3.12—vector Definitions Questionbank

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 3.12—vector Definitions with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 3.12—vector Definitions and mirrors Paper 1, 2, 3 style where relevant.

Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.

Paper

Level

Difficulty

Question 1HLPaper 1

Let a = (2, - 1, 3), b = (k, 4, - 2), c = (1, 2, 1), where k \in R . Define the scalar triple product by [a, b, c] = a \cdot (b \times c) . \dots

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Question

HLPaper 1Standard

The diagram shows a parallelogram $ABCD$ with $A(1,2,3)$ , $B(6,4,4)$ and .

(a)

(a) Find $\overrightarrow{AB}$ .

[2]

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(b)

[1]

(c)

[2]

(d)

[2]

HLPaper 1Demanding

(a)

[3]

(b)

[4]

(c)

[4]

(d)

[3]

HLPaper 1Demanding

(a)

[2]

(b)

[5]

(c)

[4]

(d)

[3]

HLPaper 1

(a)

[4]

D (2, 5, 5)

D(2,5,5)

D (2, 5, 5)

2

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Question 3

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Question 4

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Question 5

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(b) Find $\overrightarrow{AD}$ .

(c) Hence find $\overrightarrow{AC}$ .

(d) Find the coordinates of point $C$ .

Let $\vec{a} = 3\vec{i} - \vec{j} + 2\vec{k}$ and $\vec{b} = \vec{i} + 2\vec{j} - 2\vec{k}$ .

(a) Find $\vec{a}\cdot\vec{b}$ and $\left|\vec{b}\right|$ .

(b) Find the value of $\lambda$ for which $\vec{a} - \lambda\vec{b}$ is perpendicular to $\vec{b}$ .

(c) For this value of $\lambda$ , find $\left|\vec{a} - \lambda\vec{b}\right|$ in exact form.

(d) Find the exact value of $\cos\theta$ , where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ .

Let $\vec{a} = \vec{i} + 2\vec{j} - \vec{k}$ and $\vec{b} = 2\vec{i} - \vec{j} + 3\vec{k}$ .

(a) Show that $\vec{a}$ and $\vec{b}$ are not parallel.

(b) Find the values of $\lambda$ and $\mu$ for which $\lambda\vec{a} + \mu\vec{b} = 4\vec{i} + 3\vec{j} + \vec{k}$ .

(c) Show that there are no real numbers $\lambda$ and $\mu$ for which $\lambda\vec{a} + \mu\vec{b} = \vec{i} + \vec{j} + \vec{k}$ .

(d) Find the value of $p$ for which $p\,\vec{i} + \vec{j} + \vec{k}$ can be written as $\lambda\vec{a} + \mu\vec{b}$ for some real numbers $\lambda$ and $\mu$ .

Find the vector from $A$ to the point dividing $\overrightarrow{AB}$ in the ratio $2:1$ (start at $A$ ) for $A=(1,2,3)$ and $\vec{b} = 2\vec{i} - \vec{j} + 3\vec{k}$ .

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2j−

2k

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