Questions and Answers
What is a unit vector?
The unit vectors î, Ĵ, and k̂ are directed along the x, y, and z axes, respectively.
True
What is the standard Cartesian form of representing a vector F in terms of its components?
F = Fx î + Fy Ĵ + Fz k̂
The unit vector that characterizes the direction of a force can be obtained by dividing the vector d by its magnitude d, resulting in the unit vector _____ .
Signup and view all the answers
Match the following unit vectors with their corresponding axes:
Signup and view all the answers
Which of the following statements about unit vectors is correct?
Signup and view all the answers
Multiplying a unit vector by a scalar will change its direction but not its magnitude.
Signup and view all the answers
How can you obtain the unit vector n̂ in terms of vector d?
Signup and view all the answers
Which type of vector is used to represent a force acting on a specific particle?
Signup and view all the answers
The cross product of two vectors that point in opposite directions is zero.
Signup and view all the answers
What is the primary characteristic that differentiates a free vector from other types of vectors?
Signup and view all the answers
A vector is represented by the notation ____, which indicates its direction.
Signup and view all the answers
Match each type of vector to its description:
Signup and view all the answers
What is the notation typically used to express the magnitude of a vector?
Signup and view all the answers
Two vectors that are perpendicular will have a non-zero cross product.
Signup and view all the answers
Define what a sliding vector is in terms of its application.
Signup and view all the answers
What are the rectangular components of a force vector known as?
Signup and view all the answers
The orientation of the x- and y-axes affects the negativities of the components Fx and Fy.
Signup and view all the answers
What is the formula to calculate the horizontal component (Fx) of a force based on its angle?
Signup and view all the answers
The vertical component of a force can be calculated using the formula ___ = Fsinθ.
Signup and view all the answers
Which of the following statements about the rectangular components of a force is true?
Signup and view all the answers
Match the force components with their corresponding formulas:
Signup and view all the answers
When both Fx and Fy are directed to the negative axes, they are still considered positive.
Signup and view all the answers
What information do the rectangular components of a force provide about that force?
Signup and view all the answers
Study Notes
Unit Vectors
- A unit vector has a magnitude of one and points in a specified direction.
- Multiplying a unit vector by a scalar results in a vector with the unit vector's direction and magnitude equal to the scalar.
- Standard unit vectors include:
- î: directed along the x-axis
- Ĵ: directed along the y-axis
- k̂: directed along the z-axis
- n̂: an arbitrary direction
Force Representation
- A force vector F can be expressed in Cartesian coordinates as:
- 𝐅 = 𝐹𝑥 𝐢̂ + 𝐹𝑦 𝐣̂ + 𝐹𝑧 𝐤
- The unit vector n̂ that defines the direction of force F is determined by dividing the vector d by its magnitude.
Vector Notation
- Vectors distinguish between magnitude and direction via notation:
- Represented as bold (F), with an arrow above (𝐹⃗ ), or underlined (F).
- Magnitude denoted by |F|.
Types of Vectors
- Fixed vector: Represents a force with a specific point of application.
- Sliding vector: Can move along the lines of action of forces acting on rigid bodies.
- Free vector: Independent of the point of application.
Rectangular Components
- Commonly used to resolve force vectors in two dimensions:
- Components can be calculated as:
- 𝐹𝑥 = 𝐹𝑐𝑜𝑠𝜃
- 𝐹𝑦 = 𝐹𝑠𝑖𝑛𝜃
- Components can be calculated as:
- Components Fx and Fy are positive in the direction of their respective axes, negative when directed oppositely.
Direction and Specification
- Knowing the rectangular components of a force completely determines its magnitude, inclination, and direction.
- The orientation of the x- and y-axes is arbitrary, thereby affecting the sign of the components.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz covers the concept of unit vectors, their definition, and their relationship with scalar multiplication. Understand the direction and magnitude of unit vectors along the x, y, and z axes. Test your knowledge on the foundational principles of vector analysis.
