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 ↓


Representation of Vectors


Vectors are fundamental units in both math and physics, used to represent quantities that have both magnitude and direction. This makes them incredibly versatile, as they can describe anything from velocity and force in physics to points and lines in geometry. In this discussion, we are going to explore how we can represent vectors in different ways and understand their importance in vectors and matrices, especially in the context of grade 11 math. By the end of this exploration, you will be familiar with the ways vectors are used in geometry and physics, learn to perform basic operations on them, and understand their applications.

What is a vector?

A vector can be thought of as a directed line segment. It is characterized by two main properties:

  • Magnitude: The length of the vector.
  • Direction: The orientation of a vector in space.

Vectors are often denoted by letters such as a, b, or v and can be expressed in component form as v = (v1, v2, ..., vn), where each component represents the projection along the corresponding axis.

Visualization of vectors

To understand vectors better, let's visualize them. Suppose we have a two-dimensional vector v represented as follows:

<svg width="200" height="200" xmlns="http://www.w3.org/2000/svg"> <line x1="100" y1="100" x2="160" y2="60" stroke="blue" stroke-width="2"/> <circle cx="160" cy="60" r="3" fill="red" /> <line x1="100" y1="100" x2="160" y2="100" stroke="black" stroke-width="1" stroke-dasharray="4"/> <line x1="160" y1="100" x2="160" y2="60" stroke="black" stroke-width="1" stroke-dasharray="4"/> <text x="120" y="80" fill="black"><b>v</b></text> <text x="165" y="60" fill="black">(6, -4)</text> </svg>

In this vector diagram, the blue arrow represents the vector v. The vector starts at the origin (100, 100) and ends at (160, 60) in the visual grid space, which corresponds to the mathematical vector v = (6, -4) in 2D Cartesian coordinates.

Vector notation

Vectors can be represented in different notations. The most common notations are:

  • Column vector notation: A vector is written in column form:
    <vector> 6 -4 </vector>
  • Row vector notation: A vector can also be expressed horizontally:
    <vector>(6, -4)</vector>
  • Unit vector notation: When representing vectors in terms of unit vectors i and j in two dimensions:
    <vector>6 i - 4 j </vector>

Operations on vectors

Vectors can be manipulated using various operations. Here are some basic operations you can perform with vectors:

Addition of vectors

Adding vectors involves adding corresponding components. For any two vectors a and b:

<vector> a = (a1, a2)</vector>
<vector> b = (b1, b2)</vector>

Their sum a + b:

<vector> a + b = (a1 + b1, a2 + b2)</vector>

For example, if a = (2, 3) and b = (4, -1), then:

<vector> a + b = (2 + 4, 3 - 1) = (6, 2)</vector>

Subtraction of vectors

Subtracting vectors involves subtracting the components of one vector from the components of another vector. For vectors a and b:

<vector> a - b = (a1 - b1, a2 - b2)</vector>

For example, if a = (5, 7) and b = (1, 3), then:

<vector> a - b = (5 - 1, 7 - 3) = (4, 4)</vector>

Scalar multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a number). For a scalar k and a vector v:

<vector>k v = (k * v1, k * v2)</vector>

For example, if k = 3 and v = (2, -4), then:

<vector>3 v = (3 * 2, 3 * -4) = (6, -12)</vector>

Magnitude of a vector

The magnitude or length of the vector v = (v1, v2) is calculated as:

| v | = √(v1² + v2²)

For example, the magnitude for the vector v = (3, 4) is:

| v | = √(3² + 4²) = √(9 + 16) = 5

Direction of vectors

The direction of the vector relative to the positive x-axis can be determined using trigonometry. Specifically, for the vector v = (v1, v2), the angle θ is found using the tangent inverse function:

θ = tan-1 (v2/v1)

For example, if v = (3, 3), then:

θ = tan-1 (3/3) = tan-1 (1) ≈ 45°

Applications in geometry and physics

Vectors are widely used in many fields, including geometry and physics. Here are some examples:

Geometry

  • Position vectors: These vectors represent the position of a point relative to the origin. If the coordinates of point P are (x, y), then the position vector OP can be written as:
    OP = (x, y)
  • Vector representation of lines: A line can be represented by a vector, making tasks such as parallelism, perpendicularity, and finding points of intersection easier.

Physics

  • Velocity: Represents the rate of change of position with respect to time, having both magnitude (speed) and direction.
  • Acceleration: The rate of change of velocity with time, also a vector quantity.
  • Forces: Are vector quantities because they affect objects with both magnitude and direction.

Conclusion

Vectors are an essential part of math and science, especially in higher grades of study such as Grade 11. It is important for students to understand how to represent, manipulate, and apply vectors as they explore more complex problems in math, physics, and engineering. Through this discussion, we have covered the basic representations, operations, and applications of vectors, which lays a strong foundation for future studies. With this knowledge, you are now better equipped to interpret and use vectors in a variety of problems and scenarios.


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Representation of Vectors

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