Glossary
Differentiating vector-valued functions
The process of finding the derivative of a vector-valued function by differentiating each component function with respect to the independent variable.
Example:
To find the velocity vector from a position vector r(t) = <t², sin(t)>, you would perform differentiating vector-valued functions to get r'(t) = <2t, cos(t)>.
Direction (of a vector)
The orientation of a vector in space, typically measured as an angle relative to a reference axis.
Example:
If a boat is traveling due east, its direction vector would point along the positive x-axis.
Head (of a vector)
The endpoint of a vector, indicated by an arrowhead, which shows the direction of the vector.
Example:
If a vector represents a journey from point A to point B, point B would be the head of the vector.
Horizontal component (of a vector)
The projection of a vector onto the x-axis, representing its movement or influence in the horizontal direction.
Example:
For a baseball hit with an initial velocity of <80, 60> mph, 80 mph is its horizontal component of velocity.
Magnitude (of a vector)
The length or 'size' of a vector, calculated using the Pythagorean theorem from its horizontal and vertical components.
Example:
The magnitude of a force vector of <3, 4> Newtons is 5 Newtons, representing the total strength of the force.
Position vector
A vector-valued function that describes the location of a particle or object in space at a given time t.
Example:
If a car's location is given by r(t) = <3t, t²+1>, then r(t) is its position vector.
Tail (of a vector)
The starting point or origin of a vector, often assumed to be at (0,0) unless otherwise specified.
Example:
When drawing a displacement vector from your home to school, your home would be the tail of the vector.
Vector
A quantity that possesses both magnitude (length) and direction, often represented graphically as an arrow.
Example:
When a drone flies 10 meters northeast, its movement can be described by a vector showing both the distance (magnitude) and the path (direction).
Vector-valued function
A function that takes a scalar input (often time, t) and outputs a vector, typically in the form r(t) = <f(t), g(t)>.
Example:
The path of a projectile can be modeled by a vector-valued function r(t) = <v₀t cos(θ), v₀t sin(θ) - ½gt²>.
Velocity vector
The first derivative of a position vector, representing the instantaneous rate of change of position and indicating both speed and direction of motion.
Example:
For a particle moving along r(t) = <cos(t), sin(t)>, its velocity vector is v(t) = <-sin(t), cos(t)>.
Vertical component (of a vector)
The projection of a vector onto the y-axis, representing its movement or influence in the vertical direction.
Example:
For a baseball hit with an initial velocity of <80, 60> mph, 60 mph is its vertical component of velocity.