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Questions and Answers
What is a vector?
What is a vector?
A quantity with both magnitude and direction.
Define an algebraic vector.
Define an algebraic vector.
v = < a, b > where a and b are components of vector v.
What is a component of a vector?
What is a component of a vector?
One of two parts of a vector.
What is the horizontal component of a vector?
What is the horizontal component of a vector?
What is the vertical component of a vector?
What is the vertical component of a vector?
What is a scientific vector?
What is a scientific vector?
Define the magnitude of a vector.
Define the magnitude of a vector.
What is a unit vector?
What is a unit vector?
What is a zero vector?
What is a zero vector?
What is a line segment?
What is a line segment?
What is a directed line segment?
What is a directed line segment?
Define a position vector.
Define a position vector.
What is a velocity vector?
What is a velocity vector?
Define a force vector.
Define a force vector.
What is a direction angle?
What is a direction angle?
How can you find a vector from its direction and magnitude?
How can you find a vector from its direction and magnitude?
What are scalars in relation to vectors?
What are scalars in relation to vectors?
What is a scalar multiple?
What is a scalar multiple?
What is the addition of vectors?
What is the addition of vectors?
How is the subtraction of vectors defined?
How is the subtraction of vectors defined?
Define scalar multiplication of vectors.
Define scalar multiplication of vectors.
What is the unit vector in the direction of v?
What is the unit vector in the direction of v?
What is a dot product?
What is a dot product?
How do you find the angle between vectors?
How do you find the angle between vectors?
What does orthogonal mean in vector terms?
What does orthogonal mean in vector terms?
How do you determine if vectors are orthogonal?
How do you determine if vectors are orthogonal?
What is decomposition in vector terms?
What is decomposition in vector terms?
Define vector projection.
Define vector projection.
What is work in physics as it relates to vectors?
What is work in physics as it relates to vectors?
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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.
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