🔺Trigonometry Unit 11 – Vectors and Applications

What Are Vectors?

• Vectors are mathematical objects that have both magnitude and direction
• Magnitude represents the size or length of the vector, while direction indicates where the vector points in space
• Vectors are commonly represented by arrows, with the length of the arrow corresponding to the magnitude and the arrowhead indicating the direction
• Vectors can be used to describe physical quantities such as force, velocity, and displacement
• Vectors are essential in many fields, including physics, engineering, and computer graphics
• Unlike scalars, which only have magnitude, vectors provide more complete information about a quantity
• Vectors can be added, subtracted, and multiplied by scalars, allowing for various mathematical operations

Vector Notation and Components

• Vectors are typically denoted using boldface letters (e.g., $\vec{a}$) or letters with arrows above them (e.g., $\vec{v}$)
• A vector can be represented using its components, which are the projections of the vector onto the coordinate axes
• In a 2D coordinate system, a vector $\vec{v}$ can be written as $\vec{v} = (v_x, v_y)$, where $v_x$ and $v_y$ are the x and y components, respectively
  • For example, if $\vec{v} = (3, 4)$, then $v_x = 3$ and $v_y = 4$
• In a 3D coordinate system, a vector $\vec{v}$ can be written as $\vec{v} = (v_x, v_y, v_z)$, where $v_x$, $v_y$, and $v_z$ are the x, y, and z components, respectively
• The magnitude of a vector $\vec{v} = (v_x, v_y)$ can be calculated using the Pythagorean theorem: $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$
• For a 3D vector $\vec{v} = (v_x, v_y, v_z)$, the magnitude is given by $|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

Vector Operations

• Vector addition can be performed by adding the corresponding components of the vectors
  • For example, if $\vec{a} = (a_x, a_y)$ and $\vec{b} = (b_x, b_y)$, then $\vec{a} + \vec{b} = (a_x + b_x, a_y + b_y)$
• Vector subtraction is similar to addition, but the components of the second vector are subtracted from the first
  • For example, if $\vec{a} = (a_x, a_y)$ and $\vec{b} = (b_x, b_y)$, then $\vec{a} - \vec{b} = (a_x - b_x, a_y - b_y)$
• Scalar multiplication involves multiplying each component of a vector by a scalar value
  • For example, if $\vec{v} = (v_x, v_y)$ and $c$ is a scalar, then $c\vec{v} = (cv_x, cv_y)$
• The dot product (scalar product) of two vectors $\vec{a}$ and $\vec{b}$ is defined as $\vec{a} \cdot \vec{b} = a_xb_x + a_yb_y$ (for 2D vectors) or $\vec{a} \cdot \vec{b} = a_xb_x + a_yb_y + a_zb_z$ (for 3D vectors)
• The cross product (vector product) of two 3D vectors $\vec{a}$ and $\vec{b}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$, given by $\vec{a} \times \vec{b} = (a_yb_z - a_zb_y, a_zb_x - a_xb_z, a_xb_y - a_yb_x)$

Unit Vectors and Direction

• A unit vector is a vector with a magnitude of 1
• Unit vectors are used to represent direction without considering magnitude
• In a 2D coordinate system, the standard unit vectors are $\hat{i}$ (pointing along the positive x-axis) and $\hat{j}$ (pointing along the positive y-axis)
• In a 3D coordinate system, the standard unit vectors are $\hat{i}$ (x-axis), $\hat{j}$ (y-axis), and $\hat{k}$ (z-axis)
• Any vector can be expressed as a linear combination of unit vectors
  • For example, $\vec{v} = (v_x, v_y)$ can be written as $\vec{v} = v_x\hat{i} + v_y\hat{j}$
• The direction of a vector can be described using angles or direction cosines
  • Direction cosines are the cosines of the angles between the vector and the coordinate axes
• To find the unit vector in the same direction as a given vector $\vec{v}$, divide the vector by its magnitude: $\hat{v} = \frac{\vec{v}}{|\vec{v}|}$

Vector Applications in Physics

• Vectors are used extensively in physics to describe various quantities and phenomena
• Displacement is a vector that represents the change in position of an object
  • It is calculated by subtracting the initial position vector from the final position vector
• Velocity is a vector that represents the rate of change of displacement with respect to time
  • Average velocity is calculated by dividing the displacement vector by the time interval
• Acceleration is a vector that represents the rate of change of velocity with respect to time
  • It is calculated by dividing the change in velocity vector by the time interval
• Force is a vector quantity that represents the push or pull acting on an object
  • Newton's second law states that the net force on an object is equal to the product of its mass and acceleration: $\vec{F} = m\vec{a}$
• Momentum is a vector quantity defined as the product of an object's mass and velocity: $\vec{p} = m\vec{v}$
  • The conservation of momentum states that the total momentum of a closed system remains constant

Vectors in Trigonometry

• Vectors and trigonometry are closely related, as trigonometric functions are used to analyze vector components and angles
• The magnitude of a 2D vector $\vec{v} = (v_x, v_y)$ can be found using the Pythagorean theorem: $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$
• The angle $\theta$ between a vector $\vec{v}$ and the positive x-axis can be found using the arctangent function: $\theta = \tan^{-1}(\frac{v_y}{v_x})$
• The x and y components of a vector can be expressed using trigonometric functions: $v_x = |\vec{v}|\cos\theta$ and $v_y = |\vec{v}|\sin\theta$
• The dot product of two vectors $\vec{a}$ and $\vec{b}$ can be expressed using the magnitudes and the angle $\theta$ between them: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$
• The cross product of two 3D vectors $\vec{a}$ and $\vec{b}$ can be expressed using the magnitudes and the sine of the angle $\theta$ between them: $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$

Problem-Solving Strategies

• When solving vector problems, it is essential to break down the problem into smaller, manageable steps
• Identify the given information, such as vector components, magnitudes, or angles
• Determine the desired quantity or quantities to be found
• Choose an appropriate coordinate system and establish the positive directions for the axes
• Draw a diagram to visualize the problem, including vectors and any relevant angles or distances
• Apply vector operations, trigonometric functions, or other relevant concepts to solve for the unknown quantities
• Double-check your results for consistency and reasonableness
• Practice solving a variety of vector problems to develop proficiency and understanding

Real-World Vector Examples

• Wind velocity can be represented as a vector, with the magnitude indicating wind speed and the direction indicating the wind's heading
• The force acting on a sailboat is a combination of the wind force vector and the water resistance force vector
• In navigation, a ship's velocity vector can be determined by the sum of the water current vector and the ship's velocity relative to the water
• The lift force acting on an airplane wing is perpendicular to the wing's velocity vector and is crucial for maintaining flight
• In electric fields, the electric field strength at a point is represented by a vector, with the magnitude indicating the field strength and the direction indicating the force direction on a positive test charge
• Magnetic fields can be represented by vectors, with the magnetic field strength and direction indicated by the vector's magnitude and orientation
• In computer graphics, vectors are used to represent the positions, velocities, and accelerations of objects in virtual environments, enabling realistic animations and simulations