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 ↓


Cross Product


Vectors are fundamental in mathematics, especially in physics, engineering, and computer science. They help represent quantities that have both magnitude and direction, such as force, velocity, and others. In this extensive exploration, we will focus on one important vector operation: the cross product.

What is a cross product?

The cross product, also known as the vector product, is an operation on two vectors in three-dimensional space. The result of the cross product is a third vector that is perpendicular to the plane formed by the original two vectors. The cross product is usually represented as **A** × **B**, where **A** and **B** are the original vectors.

Defining vectors

Before delving deeper into the cross product, it's important to understand what vectors are:

  • A vector can be represented in component form as **A** = (Ax, Ay, Az).
  • Each component represents the projection of the vector onto the x, y, and z axes, respectively.

Similarly, another vector **B** can be represented as **B** = (Bx, By, Bz)

Calculating the cross product

The cross product of two vectors **A** and **B** is given by:

**A** × **B** = (AyBz - AzBy)**i** - (AxBz - AzBx)**j**   (AxBy - AyBx)**k**

Here, **i**, **j**, and **k** are unit vectors along the x, y, and z axes.

Example of cross product

Let us take two vectors:

  • **A** = (1, 2, 3)
  • **B** = (4, 5, 6)

To find **A** × **B**, we use the formula:

**A** × **B** = (2×6 - 3×5)**i** - (1×6 - 3×4)**j**   (1×5 - 2×4)**k**

Calculation of each component:

  • For **i** component: 2×6 - 3×5 = 12 - 15 = -3
  • For **j** component: 1×6 - 3×4 = 6 - 12 = -6
  • For **k** component: 1×5 - 2×4 = 5 - 8 = -3

Thus, the cross product becomes **A** × **B**:

(-3, -6, -3)

Geometrical interpretation

The vector obtained from the cross product is perpendicular to the original vectors **A** and **B**. This means that if you imagine both vectors lying in a plane, the cross product points straight out of the plane.

ABA × B

The red line represents the vector **A**, the blue line is the vector **B**, and the green line symbolizes the cross product **A** × **B**, which is perpendicular to both.

Properties of the cross product

Non-interchangeable

The cross product is non-commutative, which means **A** × **B** ≠ **B** × **A**. In fact, **A** × **B** = -(**B** × **A**), which shows that they are opposite vectors.

Sum over distributions

The cross product is distributive over the vector sum:

**A** × (**B**   **C**) = (**A** × **B**)   (**A** × **C**)

Orthogonal vector

The result of the cross product **A** × **B** will always be orthogonal to both **A** and **B**.

Application example

Let us explore some scenarios to understand the practical applications of cross products.

Example 1: Force on a lever

Imagine a wrench being used to turn a bolt. The force applied and the position vector from the pivot point to the point where the force is applied can be represented as vectors. Torque (twisting force) is given by the cross product of these two vectors. Torque tells us how effectively the force is rotating the object.

Example 2: Magnetic force

In physics, the force acting on a charged particle moving in a magnetic field can be determined through the cross product. Here, the velocity of the particle and the magnetic field are vectors. The force, another vector, is perpendicular to both and determines the direction and magnitude of the particle's motion.

Example 3: Rotational motion

In rotational dynamics, the rotational motion of an object can be studied using the angular velocity vector and the radius vector. Their cross product allows understanding the linear velocity, which affects rotational speeds and directions.

Conclusion

The cross product is an essential concept in mathematics, physics, and engineering. By providing a vector perpendicular to two given vectors, it helps in understanding and analyzing situations involving rotational effects, forces, and motion. Its applications range from simple geometric interpretations to complex real-world scenarios, showing its versatility and importance in a variety of fields.


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Cross Product

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