The topic focuses on the application of vectors, covering various fundamental concepts and operations involving vectors in a three-dimensional plane. Direction cosines are introduced, which define the angles a vector makes with the x, y, and z axes.
A Fish Mind Map of Unit 8: The Application of Vectors
Direction Cosines
Definition: They are the angles that a vector in a 3D plane makes with each positive (x,y,z) axis.
Definition
Definition: Thought as a shadow, it can be formed by drawing a perpendicular line from the head of one vector to another vector.
Vector
(in reference to the definition's diagram)
Scalar
where theta is the angle between vector a and vector b (in reference to the definition's diagram).
Physics Concepts
Velocity
Definition: Velocity is a vector quantity, measured by by distance travelled over time elapsed, where there's a speed (magnitude) and direction.
Forces
Definition: A force causes an object to undergo acceleration, where the magnitude of a force is measured in newtons (N).
Vector Operations
Special Cases
Area of a Parallelogram/Triangle
Formula for area of a parallelogram. Area of a triangle is 1/2 of the area of a parallelogram.
Parallel Vectors
Scalar Law
Distributive Law
Not Commutative
The cross product is a vector product
Physical Applications: Torque
Definition: Torque is a vector quantity measured in Newton-metres (N-m) or in joules (J). It's a measure of the force that can cause an object to rotate about an axis.
Properties
Magnitudes Property
Distributive Property
Associative Property With A Scalar
Commutative Property
The dot product is a scalar product
Formulas
Special Case
Perpendicular Vectors
Physical Applications: Work
Definition: when a force acting on an object causes a displacement of an object from one position to another.
Formula
Vector f is the force acting on the object, measured in N, while vector s is the displacement of the object, measured in m, making work done in J.
PROJECTIONS
Resolve
Definition: Taking a single force and decomposing it into two components. A
vector can be resolved into its corresponding horizontal and vertical components by creating a right triangle with the given vector. The magnitudes of the vertical and horizontal components can be found using primary trigonometric ratios and a given angle.
Equilibrium
It is the counterbalance of the resultant force.
Definition: a state of rest or a state of uniform motion, meaning the net force is 0, velocity is unchanging (steady speed), meaning an acceleration of 0.
