Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Vectors and Matrices ↓


Vectors


Welcome to the journey into the world of vectors. Vectors are fascinating mathematical objects that have both magnitude and direction. They are essential in mathematics and have many applications in physics, engineering, and computer science. This lesson will help you understand what vectors are, how we represent them, and how they can be used in practical situations. Don't worry if you're new to this; we'll start with the basics and move forward!

What is a vector?

In simple terms, a vector is an entity that has two main properties:

  • Magnitude: This tells us how long or large the vector is.
  • Direction: It indicates the direction the vector is pointing.

You can think of a vector as an arrow. The length of the arrow represents its magnitude, while the way it points represents its direction. Vectors can exist in one, two, or three dimensions. For simplicity, we will often use two-dimensional vectors in our examples.

Representation of vectors

The most common way to represent vectors is using coordinates. In a two-dimensional plane, a vector can be expressed as an ordered pair (x, y), where x and y are the components of the vector. For example, a vector v can be written as:

v = (3, 4)

Here, the vector v has x component of 3 and y component of 4. Let's visualize this:

(3, 4) 3 4

In this illustration, we have plotted the vector (3, 4). The arrow starts at the origin (0, 0) and points to the coordinate (3, 4).

Magnitude of a vector

The magnitude (or length) of a vector is a measure of how long it is. For a two-dimensional vector v = (x, y), the magnitude |v| can be calculated using the Pythagorean Theorem:

|v| = √(x² + y²)

This formula comes from considering the vector as the hypotenuse of a right triangle with legs x and y. Let's calculate the magnitude of the vector v = (3, 4):

|v| = √(3² + 4²) = √(9 + 16) = √25 = 5

The magnitude of v is 5. This means that if we want to measure the length of the vector directly then its length will be 5 units.

Direction of the vector

The direction of a vector tells us what angle it makes with the reference direction, usually the positive x-axis. The direction θ of a vector v = (x, y) can be determined using trigonometry. Specifically, we use the tangent function:

tan(θ) = y / x

To find θ, we use the inverse tangent (arctan or tan-1) function:

θ = arctan(y / x)

This will give us the direction angle in radians or degrees. Let's find the direction of our vector v = (3, 4):

θ = arctan(4 / 3) ≈ 53.13° (degrees)

Therefore, the vector points at an angle of approximately 53.13 degrees relative to the positive x-axis.

Types of vectors

In mathematics, we encounter different types of vectors depending on their properties and the way they are used. Here are some important types:

  • Zero vector: A vector with zero magnitude and no specific direction. It is represented as (0, 0).
  • Unit vector: A vector that has a magnitude of 1. It is often used to indicate direction only.
  • Position vector: A vector that extends from the origin to a specific point in space. It is used to represent the location of a point.
  • Equal vectors: Two vectors that have the same magnitude and direction, regardless of their initial point.
  • Negative vector: A vector that has the same magnitude as another vector, but points in the opposite direction.

Operations on vectors

Vectors can also be added, subtracted and multiplied. Let's look at these operations in detail:

Addition of vectors

The addition of two vectors consists of adding their corresponding components. If we have two vectors v = (x1, y1) and w = (x2, y2), their sum v + w is given by:

v + w = (x1 + x2, y1 + y2)

For example, if v = (3, 4) and w = (1, 2), then:

v + w = (3 + 1, 4 + 2) = (4, 6)

Imagine this addition:

(3, 4) (4, 6) (1, 2)

In this illustration, addition is essentially putting the tail of w onto the head of v. The resulting vector, in red, is (4, 6).

Subtraction of vectors

Subtracting vectors is the same as adding. To find the difference between two vectors v = (x1, y1) and w = (x2, y2), use:

v - w = (x1 - x2, y1 - y2)

Following the same vectors v = (3, 4) and w = (1, 2), the subtraction is as follows:

v - w = (3 - 1, 4 - 2) = (2, 2)

Multiplication of vectors by a scalar

Multiplying a vector by a scalar involves multiplying each component of the vector by that scalar. For a vector v = (x, y) and a scalar k, the product is:

k * v = (k * x, k * y)

If v = (3, 4) and k = 2, then:

k * v = (2 * 3, 2 * 4) = (6, 8)

This operation stretches or compresses the vector depending on the value of the scalar.

Applications of vectors

Vectors are extremely useful in real-world applications. Here are some examples:

  • Physics: Vectors are used to represent quantities such as velocity, force, and acceleration. They help in understanding these quantities visually and solving related problems.
  • Engineering: In structural analysis, vectors help determine forces and resulting stresses on materials.
  • Computer graphics: Vectors are essential in rendering images and animations, simulating motion, and creating 3D models.

Let's consider a simple example in physics. Imagine a car is traveling east at 60 km/h and then turns north at the same speed. You can represent these speeds as vectors and use vector addition to determine the resulting path of the car.

Conclusion

We have covered the basic concepts of vectors, from understanding their properties to performing operations and looking at some applications. Vectors are powerful tools in the mathematical toolbox, facilitating problem-solving in a variety of fields. The beauty and adaptability of vectors make them indispensable in a variety of mathematical topics and practical applications alike.


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