Questions and Answers
What is a vector?
A quantity with both magnitude and direction.
Define an algebraic vector.
v = < a, b > where a and b are components of vector v.
What is a component of a vector?
One of two parts of a vector.
What is the horizontal component of a vector?
Signup and view all the answers
What is the vertical component of a vector?
Signup and view all the answers
What is a scientific vector?
Signup and view all the answers
Define the magnitude of a vector.
Signup and view all the answers
What is a unit vector?
Signup and view all the answers
What is a zero vector?
Signup and view all the answers
What is a line segment?
Signup and view all the answers
What is a directed line segment?
Signup and view all the answers
Define a position vector.
Signup and view all the answers
What is a velocity vector?
Signup and view all the answers
Define a force vector.
Signup and view all the answers
What is a direction angle?
Signup and view all the answers
How can you find a vector from its direction and magnitude?
Signup and view all the answers
What are scalars in relation to vectors?
Signup and view all the answers
What is a scalar multiple?
Signup and view all the answers
What is the addition of vectors?
Signup and view all the answers
How is the subtraction of vectors defined?
Signup and view all the answers
Define scalar multiplication of vectors.
Signup and view all the answers
What is the unit vector in the direction of v?
Signup and view all the answers
What is a dot product?
Signup and view all the answers
How do you find the angle between vectors?
Signup and view all the answers
What does orthogonal mean in vector terms?
Signup and view all the answers
How do you determine if vectors are orthogonal?
Signup and view all the answers
What is decomposition in vector terms?
Signup and view all the answers
Define vector projection.
Signup and view all the answers
What is work in physics as it relates to vectors?
Signup and view all the answers
Study Notes
Vectors Overview
- Vectors have both magnitude (length) and direction.
- Algebraic vector representation: v = < a, b >.
- Components of a vector are its coordinates, indicating its position in space.
Components of Vectors
- Horizontal component: First part of a vector representing the x-coordinate.
- Vertical component: Second part of a vector representing the y-coordinate.
Scientific Representation
- Scientific vector form: v = ai + bj, where i and j denote unit vectors along the axes.
Magnitude and Properties
- Magnitude of vector v = < a, b > calculated as ‖v‖ = √(a² + b²).
- Unit vector: A vector with a magnitude of 1.
- Zero vector: A vector with zero magnitude, lacking direction.
Basic Vector Definitions
- Directed line segment: Line segment from point P to Q, defining a geometric vector.
- Position vector: Origin at (0,0) and terminal point at (a,b).
Motion and Forces
- Velocity vector: Indicates speed and direction of an object.
- Force vector: Represents force's direction and magnitude acting on an object.
Angular and Directional Concepts
- Direction angle (α): Angle formed between the position vector and x-axis, ranging from 0° to 360°.
Constructing Vectors
- To find a vector from its magnitude and angle: v = ‖v‖(cos αi + sin αj).
- Scalars: Real numbers that represent quantities with only magnitude.
Vector Operations
- Scalar multiple of a vector modifies its magnitude and potentially its direction.
- Vector addition involves combining components: v + w = (a₁ + a₂)i + (b₁ + b₂)j.
- Vector subtraction is done similarly: v - w = (a₁ - a₂)i + (b₁ - b₂)j.
Unit Vectors and Projections
- Unit vector in the direction of v calculated as u = v / ‖v‖.
- Decomposition of vector v relates to orthogonal projections onto another vector w.
Dot Product and Orthogonality
- Dot product: v * w = a₁a₂ + b₁b₂, resulting in a scalar.
- Two vectors are orthogonal if their dot product equals zero.
Vector Decomposition
- To decompose vector v relative to vector w:
- Parallel component: v₁ = ((v * w)/(‖w‖²))w.
- Orthogonal component: v₂ = v - v₁.
Work Done by Forces
- Work (W) is calculated as the product of force magnitude and distance: W = (‖F‖)(‖→AB‖).
- If force is applied at an angle, the work done is W = F * →AB.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz covers the fundamental concepts of vectors, including their definitions, components, and scientific representation. You'll explore key properties such as magnitude, unit vectors, and applications in motion and force. Test your understanding of the foundational elements of vector algebra.
