Unit 8: Application of Vectors

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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.

CROSS PRODUCT

DOT PRODUCT

Unit 8: Application of Vectors