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 ↓


Vector Operations


Vectors are fascinating mathematical objects that have both magnitude and direction. They are incredibly useful in many real-world situations, from physics and engineering to computer graphics and navigation systems. In this lesson, we will explore vector operations in the context of high school math, focusing on the essential operations that can be performed on vectors: addition, subtraction, and scalar multiplication. We will use a simple and clear approach to demonstrate these operations, supported by examples and visual aids.

Understanding vector

Before we get into vector operations, let's first define what a vector actually is. A vector is a directed quantity; this means that it has both a direction and a magnitude (or size). You can think of a vector like an arrow. An arrow has a certain length (magnitude) and it points in a particular direction.

In mathematics, we often represent vectors as coordinates in a coordinate system. A vector in two-dimensional space (2D) can be represented by two numbers, say ( mathbf{v} = (x, y) ), where ( x ) and ( y ) are the components of the vector along the x-axis and y-axis, respectively.

Vector addition

Vector addition is one of the most basic operations we can perform with vectors. When we add two vectors, we essentially place them tip-to-tail and then create a new vector from the beginning of the first vector to the end of the second vector.

Example of vector sum

Suppose we have two vectors:

(mathbf{a} = (a_1, a_2)) (mathbf{b} = (b_1, b_2))
(mathbf{a} = (a_1, a_2)) (mathbf{b} = (b_1, b_2))

The sum of these two vectors, (mathbf{c} = mathbf{a} + mathbf{b}), is given by:

(mathbf{c} = (a_1 + b_1, a_2 + b_2))
(mathbf{c} = (a_1 + b_1, a_2 + b_2))

Let's imagine this addition:

A B A+B

In the above diagram, the blue vector ( mathbf{a} ) is added to the red vector ( mathbf{b} ). The resulting green vector is the vector sum ( mathbf{c} = mathbf{a} + mathbf{b} ).

Vector subtraction

Vector subtraction might be a little more intuitive if you think of subtracting a vector as adding its negative. The negative of a vector has the same magnitude, but the opposite direction. For vector subtraction, you perform a similar operation: place the two vectors tip-to-tail, but this time "add" the opposite of the vector you want to subtract.

Example of vector subtraction

Two vectors are given:

(mathbf{a} = (a_1, a_2)) (mathbf{b} = (b_1, b_2))
(mathbf{a} = (a_1, a_2)) (mathbf{b} = (b_1, b_2))

The difference, (mathbf{c} = mathbf{a} - mathbf{b}), is given by:

(mathbf{c} = (a_1 - b_1, a_2 - b_2))
(mathbf{c} = (a_1 - b_1, a_2 - b_2))

Let's visualize this operation:

B A - B

Here, the purple represents the vector ( mathbf{a} ), and the red dashed line represents the negative of ( mathbf{b} ). The solid black line represents the vector ( mathbf{a} - mathbf{b} ).

Scalar multiplication

Scalar multiplication involves multiplying a vector by a scalar, which is a fancy word for a constant number. When you multiply a vector by a scalar, it increases or decreases the magnitude of the vector without changing its direction if the scalar is positive. It also reverses the direction if the scalar is negative.

Example of scalar multiplication

Consider the vector:

(mathbf{v} = (v_1, v_2))
(mathbf{v} = (v_1, v_2))

and a scalar ( k ). The result of multiplying a vector by a scalar is:

(kmathbf{v} = (kv_1, kv_2))
(kmathbf{v} = (kv_1, kv_2))

Let's look at scalar multiplication:

V 2V

In this diagram, the red arrow represents the original vector ( mathbf{v} ), and the blue arrow represents the vector ( 2mathbf{v} ). As shown, multiplying ( mathbf{v} ) by ( 2 ) doubles its magnitude.

Properties of vector operations

Vector operations also have many interesting properties that can be very useful, especially when solving mathematical problems. Here are some of these properties:

  • Commutative property: Vector addition is commutative, meaning the order in which you add the vectors doesn't matter. 
        (mathbf{a} + mathbf{b} = mathbf{b} + mathbf{a})
  • Associative property: Vector addition is associative. This means that when three or more vectors are added, the result is the same, no matter how the vectors are grouped. 
        ((mathbf{a} + mathbf{b}) + mathbf{c} = mathbf{a} + (mathbf{b} + mathbf{c}))
  • Distributive property: Scalar multiplication distributes over vector addition. 
        (k(mathbf{a} + mathbf{b}) = kmathbf{a} + kmathbf{b})
  • Zero vector: Adding a zero vector (a vector with no magnitude) to any vector does not change the vector. 
        (mathbf{a} + mathbf{0} = mathbf{a})
  • Identity for scalar multiplication: Multiplying a vector by 1 does not change the vector. 
        (1mathbf{a} = mathbf{a})

Example problems with solutions

Let's look at some examples of using vector operations to solve problems.

Example 1: Adding vectors

Suppose you have the vectors:

(mathbf{u} = (3, 4)) (mathbf{v} = (1, 2))
(mathbf{u} = (3, 4)) (mathbf{v} = (1, 2))

Find ( mathbf{u} + mathbf{v} ).

Solution:

(mathbf{u} + mathbf{v} = (3 + 1, 4 + 2) = (4, 6))
(mathbf{u} + mathbf{v} = (3 + 1, 4 + 2) = (4, 6))

Example 2: Subtracting vectors

Using similar vectors, find ( mathbf{u} - mathbf{v} ).

Solution:

(mathbf{u} - mathbf{v} = (3 - 1, 4 - 2) = (2, 2))
(mathbf{u} - mathbf{v} = (3 - 1, 4 - 2) = (2, 2))

Example 3: Scalar multiplication

Find ( 3mathbf{u} ).

Solution:

(3mathbf{u} = (3 times 3, 3 times 4) = (9, 12))
(3mathbf{u} = (3 times 3, 3 times 4) = (9, 12))

Conclusion

Vector operations are powerful tools in mathematics. By mastering vector addition, subtraction, and scalar multiplication, you can solve complex problems involving direction and magnitude. As you have seen in this lesson, these operations are not just abstract mathematical concepts but are extremely practical in a variety of fields ranging from engineering to physics and beyond. Continue practicing these operations with different vectors to get comfortable using them, and soon, vector operations will become a second nature to you.

As your skills grow, you can consider exploring more advanced topics related to vectors, such as dot products, cross products, and their applications in three-dimensional space.


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