11.5.1: Resources and Key Concepts

Resources

Videos

• Algebraic Vectors
  • Vectors - An Algebraic Approach
• Vector Arithmetic
  • Vectors - Vector Arithmetic and Properties of Vectors
• Component Form and Unit Vectors
  • Vectors - Unit Vectors and Component Form
• Applications
  • Vectors - Applications Involving Vectors

Key Concepts

Definitions

• Vector Components: A vector \(\vec{v}\) can be broken down into a horizontal vector component, \(\vec{v}_x\), and a vertical vector component, \(\vec{v}_y\), such that \(\vec{v} = \vec{v}_x + \vec{v}_y\). The scalar values \(v_x\) and \(v_y\) are called the components of the vector.
• Unit Vector: A vector with a magnitude of 1. The unit vectors in the horizontal and vertical directions are denoted by \(\vec{i}\) and \(\vec{j}\), respectively.
• Coordinate Form (of a vector): A vector \(\vec{v}\) is in coordinate form when it is written as the sum of its scalar components multiplied by the standard unit vectors: \(\vec{v} = v_x\vec{i} + v_y\vec{j}\).
• Normalizing (a vector): The process of creating a unit vector in the same direction as a given non-zero vector by dividing the vector by its magnitude.
• Zero Vector: A vector with a magnitude of zero and an undefined direction, denoted \(\vec{0}\). It is also known as a null vector.
• Static Equilibrium: A state in which an object or particle is motionless because the sum of all forces acting on it is the zero vector.

Theorems

• Components: If \(\theta\) is the angle measured counter-clockwise from the positive x-axis to the vector \(\vec{v}\), then the scalar components of the vector are given by \(v_x = ||\vec{v}||\cos(\theta)\) and \(v_y = ||\vec{v}||\sin(\theta)\).
• Vector Component Formula: The vector components of \(\vec{v}\) are \(\vec{v}_x = ||\vec{v}||\cos(\theta)\vec{i}\) and \(\vec{v}_y = ||\vec{v}||\sin(\theta)\vec{j}\).
• Vector Magnitude Formula: For a vector \(\vec{v} = v_x\vec{i} + v_y\vec{j}\), the magnitude is \(||\vec{v}|| = \sqrt{v_x^2 + v_y^2}\) and the direction angle \(\theta\) satisfies \(\tan(\theta) = \frac{v_y}{v_x}\).
• Conversion Between Geometric and Coordinate Forms: The formulas for components and magnitude/direction allow for conversion between the geometric representation (magnitude and direction) and the coordinate form of a vector.
• Scalar Multiplication (coordinate form): If \(\vec{v} = v_x\vec{i} + v_y\vec{j}\) and \(k\) is a scalar, then \(k\vec{v} = (kv_x)\vec{i} + (kv_y)\vec{j}\).
• Unit Vector Scaling: The unit vector \(\vec{u}\) in the direction of \(\vec{v}\) is given by \(\vec{u} = \frac{1}{||\vec{v}||}\vec{v}\). A vector \(\vec{w}\) of length \(k\) in the direction of \(\vec{v}\) is given by \(\vec{w} = k\vec{u}\).
• Vector Addition (coordinate form): If \(\vec{u} = a\vec{i} + b\vec{j}\) and \(\vec{v} = c\vec{i} + d\vec{j}\), then \(\vec{u} + \vec{v} = (a+c)\vec{i} + (b+d)\vec{j}\).

Common Mistakes

• Confusing Vector Components and Components: The vector components (\(\vec{v}_x, \vec{v}_y\)) are vectors, while the components (\(v_x, v_y\)) are scalars that represent the signed lengths of the vector components.
• Using Heading as the Standard Angle \(\theta\): When finding components, the angle \(\theta\) must be the angle in standard position (measured counter-clockwise from the positive x-axis), not necessarily the heading or bearing angle given in a navigation problem.
• Incorrectly Determining the Quadrant for Direction: When finding the direction angle \(\theta\) from components using \(\arctan(\frac{v_y}{v_x})\), you must use the signs of \(v_x\) and \(v_y\) to determine the correct quadrant for \(\theta\), as the arctangent function only returns angles in Quadrants I or IV.