Vectors

Understanding Vectors

• Vectors are mathematical expressions used to represent movement or translation in space. They have magnitude (length) and direction.
• Vectors are often depicted as arrows. The length of the arrow represents the magnitude of the vector, while the arrow’s trajectory represents the direction.
• Vector values are typically written as column vectors, such as (2,3), or as coordinates with a horizontal and vertical component.
• A vector is a displacement. It details a journey from one place to another, not the points at which the journey starts or finishes.
• We often represent vectors using letters. Lowercase letters in bold or with a line underneath are commonly used.

Vector Operations

• Adding vectors: To add vectors together, add the corresponding components together. This is often referred to as the tip-to-tail method.
• Subtracting vectors: To subtract one vector from another, subtract the corresponding components. This is like adding a negative vector.
• Multiplying vectors by a scalar: To multiply a vector by a number (scalar), multiply each component by that number. This changes the magnitude of the vector but not its direction.

Vector Properties

• Equal vectors: Two vectors are equal if and only if their corresponding components are equal.
• Zero vector: This is a vector with all components equal to zero. It has a magnitude of zero and is the only vector without a precise direction.
• Negative vectors: A negative vector has the same magnitude as the original vector but points in the exact opposite direction.
• A vector multiplied by a negative number will reverse its direction but maintain its magnitude.

Vector Calculation in Geometry

• Vectors can be used to prove relationships in geometry such as parallel lines and midpoints.
• Collinear points: Points are said to be collinear if they lie on the same straight line. This can be proven using vectors.
• Vectors are often used to calculate displacement in physics as well.