‖𝑣‖ = $\sqrt{a^2 + b^2}$ where $v = <a, b>$.

$tan(θ)= \frac{\text{vertical component}}{\text{horizontal component}}$

$r'(t) = <f'(t), g'(t)>$

$v(t) = r'(t)$

$a^2 + b^2 = c^2$

horizontal component = magnitude * cos(θ)

vertical component = magnitude * sin(θ)

$\hat{u} = \frac{\vec{v}}{|\vec{v}|}$

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|cos(\theta)$

$\theta = cos^{-1}(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|})$

A quantity with both direction and magnitude.

The length of the vector.

A function that returns a vector, often written as $r(t) = <f(t), g(t)>$.

The tangent vector to the curve at a given point, representing the velocity vector.

The point at which the vector originates.

The final point of the vector, represented with an arrowhead.

The horizontal and vertical components that define the vector's direction and magnitude.

‖𝑣‖

r'(t) = <f'(t), g'(t)> where r(t) = <f(t), g(t)>

A vector that represents the location of a point in space relative to an origin.

Differentiate each component of the vector-valued function separately. If $r(t) = <f(t), g(t)>$, then $r'(t) = <f'(t), g'(t)>$.

The speed of the particle moving along the path defined by the vector-valued function.

The velocity vector is the first derivative of the position vector with respect to time: $v(t) = r'(t)$.

Velocity is the derivative of position, and acceleration is the derivative of velocity. Therefore, $v(t) = r'(t)$ and $a(t) = v'(t) = r''(t)$.

A vector-valued function traces a curve in space as the parameter t varies. Each value of t corresponds to a point on the curve, with the vector pointing from the origin to that point.

The direction is determined by the ratio of the vertical component to the horizontal component, which can be used to find the angle with respect to the horizontal axis.

The second derivative, or acceleration vector, indicates the rate of change of the velocity vector, providing information about how the speed and direction of motion are changing.

A vector-valued function can be represented by a set of parametric equations, where each component of the vector is a function of the parameter t.

The magnitude of the velocity vector, also known as speed, is the rate of change of arc length with respect to time. Integrating the speed over an interval gives the arc length traversed during that interval.

A vector-valued function is continuous if each of its component functions is continuous.