vectorfunctions Tag Posts Index

Tangential and Normal Components of Acceleration

In this video, I write the vector equation of acceleration in terms of its unit tangent and unit normal vector components, which is useful when analyzing the motion

Example 3: Integrating Acceleration and Velocity Vectors to get the Position Vector

In this video, I show that we can obtain the velocity of a moving object by first integrating its acceleration vector to obtain its velocity and then integrating

Motion in Space: Velocity and Acceleration

In this video, I go over further into vector functions, and this time look at motion in space in terms of the velocity, acceleration, and position of objects moving

Motion in Space: Velocity and Acceleration

In this video, I go over further into vector functions, and this time look at motion in space in terms of the velocity, acceleration, and position of objects moving

Osculating Circle of a Parabola + Summary of TNB Vectors and Curvature Formulas

In this video, I go over the osculating circle, which lies on the same osculating plane and has the same tangent and normal vector as a given point on a space curve.

Normal and Binormal Vectors + Example on a Circular Helix

In this video, I go over the principal unit normal and binormal vectors of a space curve, which are both derived from the unit tangent vector. Since the unit tangent

Curvature of a Plane Curve y = f(x) + Example on a Parabola

In this video, I go over the curvature of the special case of a plane curve with equation y = f(x), by selecting x as the parameter and using our earlier cross product

Curvature of a Plane Curve y = f(x) + Example on a Parabola

In this video, I go over the curvature of the special case of a plane curve with equation y = f(x), by selecting x as the parameter and using our earlier cross product

Curvature Cross Product Formula: Proof and Example on Twisted Cubic

In this video, I derive a more convenient formula for the curvature of a space curve, which involves the cross product of the first two derivatives of the vector

Arc Length of a Space Curve + Example on Circular Helix

In this video, I extend the formula for the arc length of a 2D plane curve to that of a 3D space curve, both of which can be represented in compact form as the integral

Vector Functions: Arc Length and Curvature

In this video, I go over further into vector functions and this time do a deep dive into the arc length and curvature of a space curve. I also go over normal, binormal

Example 4: Tangent Vector is Orthogonal to Vector of Constant Length

In this video I show that the tangent vector (derivative of a vector function) is orthogonal or perpendicular to a vector of constant length. Geometrically this

Differentiation Rules for Vector Functions

In this video I go over the six main differentiation rules for vector functions, as well as the 2nd (or higher) derivatives. The rules have their similar counterparts

Example 2: Sketching the Tangent Vector

In this video I go over another example on derivatives of vector functions, and this time sketch the tangent vector to a plane curve manually as well as with GeoGebra's

Derivative for Vector Functions by Components + Proof

In this video I show that the derivative of a vector function is equal to the derivative of the real-valued functions in its components. I prove this by applying

Derivative for Vector Functions by Components + Proof

In this video I show that the derivative of a vector function is equal to the derivative of the real-valued functions in its components. I prove this by applying

Definition of Derivative for Vector Functions

In this video I go over the definition of derivative for a vector function, which is very similar to the definition for real valued functions. The derivative of

Definition of Derivative for Vector Functions

In this video I go over the definition of derivative for a vector function, which is very similar to the definition for real valued functions. The derivative of

Derivatives and Integrals of Vector Functions

In this video I go further into vector functions and this time show how to obtain their derivatives and integrals. The definition of derivative for a vector function

Derivatives and Integrals of Vector Functions

In this video I go further into vector functions and this time show how to obtain their derivatives and integrals. The definition of derivative for a vector function