Statics

(5) Force vectors

Instructor:

Dr. Sawsan Alkhawaldeh

Department of Civil Engineering

Cartesian Vectors (3D)

Right-Handed Coordinate System

A rectangular coordinate system is said to be right-handed if the thumb of the right hand points in the direction of the positive z axis when the right-hand fingers are curled about this axis and directed from the positive x towards the positive y axis.

Cartesian Vectors (3D)

Cartesian vector analysis is often used to solve problems in three dimensions.

Rectangular Components of a Vector

𝐴 = 𝐴𝑥 + 𝐴𝑦 + 𝐴𝑧

Cartesian Vectors (3D)

Cartesian Unit Vectors

Remember: The sense (or arrowhead) of these vectors will be represented analytically by a plus or minus sign, depending on whether they are directed along the positive or negative x, y, or z axes.

Cartesian Vectors (3D)

Cartesian Vector Representation

Cartesian vector is represented as:

Cartesian Vectors (3D)

Magnitude of a Cartesian Vector

The magnitude of A is equal to the positive square root of the sum of the squares of its components.

Cartesian Vectors (3D) Direction of a Cartesian Vector

Cartesian Vectors (3D) Unit Vector

is a vector of length 1 unit, sometimes also called a direction vector.

Addition of Cartesian Vectors

𝑨 = 𝐴𝑥𝒊 + 𝐴𝑦𝒋 + 𝐴𝑧𝒌 𝑩 = 𝐵𝑥𝒊 + 𝐵𝑦𝒋 + 𝐵𝑧𝒌

Example (1) Express the force F as a Cartesian vector.

Example (2) Determine the magnitude and the coordinate direction angles of the resultant force acting on the ring.

Example (3) Express the force F as a Cartesian vector.

Example (4) Specify the magnitude of F2 and its coordinate direction angles so that the resultant force FR acts along the positive y axis and has a magnitude of 800 N.

Quiz (1) Three forces act on the bracket. Determine the magnitude and direction of F2 so that the resultant force is directed along the positive u axis and has a magnitude of 50 lb.

Problem (1)

Express the force as a Cartesian vector.

Problem (2) Determine the coordinate direction angles of the force.

Problem (3) Express the force as a Cartesian vector.

Problem (4)

Express the force as a Cartesian vector.

Problem (5) Determine the resultant force acting on the hook.

Dot Product Dot product of vectors A and B , written A · B, it is defined as

the product of the magnitudes of A and B and the cosine of the angle 𝜃 between their tails which expressed as:

𝐴 ∙ 𝐵 = 𝐴𝐵 cos 𝜃

Laws of Operation

Cartesian Vector Formulation

Applications of Dot Product The dot product is used to determine the angle between two vectors or the projection of a vector in a specified direction.

The angle formed between two vectors or intersecting lines,

The components of a vector parallel and perpendicular to a line,

Example (1) Determine the magnitudes of the projection of the force F onto the u and v axes.