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 ↓


Projection of Vectors


Vector projection is a topic in mathematics that involves the projection of one vector onto another. In simple terms, it helps us determine how much of one vector lies in the direction of another vector. This concept is particularly useful in physics and engineering, especially in contexts that involve force, momentum, and finding the components of a vector.

Understanding vector

A vector is a quantity that has both magnitude and direction. Vectors are usually represented as arrows in space where the length of the arrow represents the magnitude and the direction represents the direction of the vector.

In algebraic terms, the vector v can be represented as:

v = [v1, v2, ..., vn]

This is a simple notation for a vector in n-dimensional space. For simplicity, we often work with 2-dimensional or 3-dimensional space.

What is projection?

The projection of a vector u onto another vector v is a way of describing the effect of a vector v on a vector u. In essence, it is the shadow or image of the vector u onto the vector v.

Visual example

Below is a visual example of vector projection, where the vector u is being projected onto v:

You V ProseVYou

In this image, the projection of vector u onto vector v is shown as a dashed green line. It shows how much of vector u is in the direction of vector v.

The math behind the projection

The mathematical formula for the projection of a vector u onto a vector v is given by:

projv(u) = (u · v / v · v) v

Here, the point (·) represents the dot product of two vectors. The dot product of two vectors a and b is calculated as follows:

a · b = a1 b1 + a2 b2 + ... + an bn

The result of the projection is a scalar multiple of the vector v.

Step-by-step example

Let us take a very simple example to understand this concept better. Consider the vectors u = [4, 3] and v = [2, 1].

  1. Calculate the dot product u · v:
    2. u · v = 4 * 2 + 3 * 1 = 8 + 3 = 11
  2. Calculate the dot product v · v:
    3. v · v = 2 * 2 + 1 * 1 = 4 + 1 = 5
  3. Substitute these values into the projection formula:
    4. projv(u) = (11 / 5) * [2, 1]
  4. Perform scalar multiplication:
    5. projv(u) = [22/5, 11/5] which is approximately equal to [4.4, 2.2]

The vector [4.4, 2.2] represents the projection of a vector u onto a vector v, which shows how much of u points in the direction of v.

Why is vector projection useful?

Vector projection is highly useful in many fields such as physics, graphics, and engineering. For example, in physics, it is used to break down forces into components parallel and perpendicular to surfaces. This can be important when analyzing objects located on a slope or any situation involving gravity.

Projection in higher dimensions

The theory of vector projection is not limited to 2-dimensional or 3-dimensional spaces. It can be extended to n-dimensional spaces, which can be particularly useful in advanced fields such as machine learning and data science where very high-dimensional spaces are common.

More visual examples

Consider another example with different vectors a and b:

A B ProseBA

In this example, the orange line represents the vector b, the purple line represents the vector a, and the dashed blue line indicates the projection of a onto b.

Application

There are many applications of vector projection. Here are some examples:

  • Physics: Calculating the components of velocity, force, or acceleration in a specific direction.
  • Graphics: Adjusting the view in 3D graphics to correctly simulate perspective.
  • Engineering: Analyzing structural stress and strain along special lines in construction.
  • Artificial intelligence: Using high-dimensional projections to reduce dimensions for pattern recognition.

Conclusion

Understanding vector projection helps in a variety of mathematical and real-world applications. By splitting a vector into components in different directions, we can analyze and solve complex problems in a more manageable way. This concept provides valuable information about the nature of vectors and how they interact with each other in different spaces. Whether you are solving a problem in physics or calculating dimensions in graphics, vector projection is a fundamental tool that makes these tasks possible.


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Projection of Vectors

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