Vector Basics and Operations

Vector Basics

• A vector is a quantity that has both magnitude and direction.
• Notation typically includes lowercase letters (e.g., a, b) for vectors.
• Position vectors can be represented in a 3-D coordinate system using coordinates (xi, yi, zi).

Vector Operations

• Scalar multiplication of a vector a by a scalar k is defined as ka = (k * a).
• Two vectors are equal if their corresponding components are equal.
• The zero vector has a magnitude of zero and is represented as 0.

Unit Vectors

• A unit vector in the direction of a vector q' is denoted by (\hat{q'} = \frac{q'}{|q'|}).
• The magnitude of a vector can be computed using the formula (|\vec{v}| = \sqrt{x^2 + y^2 + z^2}).

Types of Vectors

• Collinear Vectors: Two vectors are collinear if they lie on the same line, regardless of their magnitudes.
• Equal Vectors: Two vectors are equal if they have the same magnitude and direction, irrespective of their initial points.
• Negative Vectors: A vector with the same magnitude as a vector a but in the opposite direction.

Vector Addition

• The triangle law of vector addition states that if A and B are two vectors, then the resultant C = A + B can be represented as the third side of a triangle.
• The parallelogram law of addition can also represent the addition of two vectors geometrically.

Dot Product

• The dot product (or scalar product) of two vectors a and b is denoted by (a \cdot b) and related to the cosine of the angle θ between them: (a \cdot b = |a||b|\cos(\theta)).
• The dot product is zero if vectors a and b are perpendicular.

Cross Product

• The cross product of two non-zero vectors a and b, denoted by (a \times b), results in a vector that is perpendicular to both a and b.
• The magnitude of the cross product is given by (|a \times b| = |a||b|\sin(\theta)).

Projection of Vectors

• The projection of vector a onto vector b is calculated using the formula: [ \text{proj}_{b}(a) = \frac{a \cdot b}{|b|^2}b ]

Cauchy-Schwarz Inequality

• For any two vectors a and b, the inequality ( |a \cdot b| \leq |a||b| ) holds true.

Velocity and Acceleration

• The velocity vector is the derivative of the position vector: (v(t) = \frac{d}{dt}r(t)).
• Acceleration is the time derivative of the velocity vector: (a(t) = \frac{d}{dt}v(t)).

Important Definitions

• Component of a Vector: The projections of a vector onto the axes of a coordinate system.
• Direction Cosines: Cosines of the angles made by a vector with the coordinate axes.

Relationships & Theorems

• If two vectors (a) and (b) are parallel, then (a \times b = \mathbf{0}) (the zero vector).
• If two vectors a and b form an angle θ, their relationship is dependent on the sine and cosine functions of that angle.

This quiz covers the fundamental concepts of vectors, including definitions, operations, and types of vectors. You will explore scalar multiplication, unit vectors, and learn how to identify collinear and equal vectors. Test your understanding of vector notation and calculations.