The Equation of a Line Perpendicular to a Vector
Finding the equation of a line perpendicular to a vector in \(R^2\)
Problem: Find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) for the following vectors:
Finding the equation of a line perpendicular to a vector in \(R^2\)
Finding the equation of a plane perpendicular to a vector in \(R^3\)
Calculating vector operations including the dot product
Finding the angle between two vectors
Find the projection of one vector onto another
Calculating a linear combination and dot product of vectors
Finding the scalar and vector projection of one vector onto another
Finding the vector and scalar projection of one vector onto another
Calculating the dot product
Finding the angle between vectors
Solving for orthogonal vectors using the dot product
Using the dot product to calculate work done
Find the measure of an angle given the three vertices for the angle
Using scalar projection to find the distance from a point to a line
Finding the scalar and vector projection of one vector onto another
Using vectors to find the work done when a force moves an object
Finding the scalar and vector projection of one vector onto another
Using vectors to prove facts about equilateral triangles and a scalar multiplication property for the dot product
Proofs of the Cauchy-Schwarz Inequality and a property of vector projections
Proving facts about the derivatives of vector functions including the product rule
Calculating slopes of tangent lines to parametric curves
Calculating the dot product for three-dimensional vectors
Calculating the dot product for three-dimensional vectors given the vectors' lengths and the angle between them
Calculating the angle between two three-dimensional vectors
Calculating the angle formed by three points in three-dimensional space
Finding the scalar and vector projection of one three-dimensional vector onto another
Finding the angle of intersection for two three-dimensional vector functions
Finding a directional derivative for a function of two variables
Finding a rate of change for a function of two variables in a given direction
Finding the maximum rate of change and the direction it occurs for a function of two variables
Finding a unit vector in the direction of a given vector
Finding the component form and magnitude for a vector between two points
Performing calculations with vectors including addition, subtraction, and scalar multiplication
Finding the equations of parallel and perpendicular lines
Finding the component form and magnitude of a vector
Adding and subtracting vectors graphically
Finding a unit vector in the same direction as a given vector
Finding the component form for a linear combination of the standard unit vectors
Finding a vector given its length and angle with the positive x-axis
Basic Vector Operations such as Addition, Subtraction, and Scalar Multiplication
Cartesian equations and parametric equations of curves
Derivatives and vectors with some physics applications
Tangent lines to parametric equations and related rates examples
Review of vector functions and parametric equations
Review of limits, continuity, and the Intermediate Value Theorem
Review of the limit definition of a derivative and calculating the derivative
Implicit differentiation and physics applications of derivatives
Derivatives of parametric equations and tangent lines
Review of the limit definition of a derivative and calculating the derivative
Antiderivatives and physics applications
Review of the limit definition of a derivative and calculating the derivative
Review of the limit definition of a derivative and calculating the derivative
Derivatives and vectors with some physics applications
Derivatives and vectors with some physics applications
Tangent lines to parametric equations and related rates examples
Finding vectors between two points
Calculating the length of a vector and finding linear combinations of vectors
Finding a vector from its length and angle with the positive x-axis
Finding the vector with a given length and direction
Using vectors and trigonometry to find the true speed and direction of a boat
Finding a vector equation for the line through two points
Finding the distance between a point and a line using scalar projection
Using vectors to find the magnitude and direction of a resultant force
Finding a line perpendicular to a vector and passing through a point
Find the Cartesian form and sketch the graph for the parametric equation of a line
Finding a unit vector
Finding a parallel vector through a point
Graphing a vector function by converting it to a Cartesian equation
Graphing a vector function by converting it to a Cartesian equation
Finding a vector equation for the line passing through two points
Basic Vector Operations such as Addition, Subtraction, and Scalar Multiplication
Proving associative and scalar multiplication properties for vectors
Using vectors to prove the midpoint formula and a fact about quadrilaterals
Proving the Triangle Inequality and a property of the midpoints of rectangles
Proving a certain parameterization gives the unit circle and showing vector equations of lines are parallel
Cartesian equations and parametric equations of curves
Derivatives and vectors with some physics applications
Tangent lines to parametric equations and related rates examples
Implicit differentiation and physics applications of derivatives
Derivatives of parametric equations and tangent lines
Review of the limit definition of a derivative and calculating the derivative
Antiderivatives and physics applications
Using the limit definition to find derivatives of functions and vector functions
Review of the limit definition of a derivative and calculating the derivative
Review of the limit definition of a derivative and calculating the derivative
Derivatives and vectors with some physics applications
Tangent lines to parametric equations and related rates examples
Finding the vector between two points in dimensional space
Calculating the length of a linear combination of vectors
Finding a vector with a specified direction and length
Finding the cross product between two three-dimensional vectors
Using the cross product to find a vector orthogonal to the plane defined by three points
Using the cross product to find the area of a parallelogram in three-dimensional space
Calculating the length of the cross product of two vectors given information about the vectors
Writing an equation for a plane through a point and perpendicular to a given line
Finding an equation of the plane containing two given lines in three-dimensional space
Finding the length of a three-dimensional curve
Finding the curvature of a three-dimensional vector function
Finding the position vector function given the velocity and an initial position
Finding the tangent plane and the normal line for a surface
Evaluating a line integral of a vector field along an curve
Evaluating a line integral of a vector field along a curve in space
Using a line integral to find the work done by a force field moving a particle along a curve
Finding the curl and divergence of a three dimensional vector field
Finding the curl and divergence of a three dimensional vector field
Evaluating a surface integral of a vector field
Using Stokes' Theorem to express a surface integral of the curl of a vector field as a single integral
Using the Divergence Theorem to find the flux of a vector field across a surface
Using the Divergence Theorem to evaluate the surface integral of a vector field
Explaining the definitions of real and complex vectors with examples
Determining if two vectors are linearly independent
Defining scalar multiplication and giving an example
Defining vector addition and giving an example
Finding a linear combination of vector functions
Explaining the concepts of existence and uniqueness for solutions of systems of equations
Defining linear independence and dependence for vectors
Explaining two vectors are linearly dependent if and only if they are scalar multiples
Showing that three vectors are linearly independent
Using linear independence to show the uniqueness of a vector equation solution
Defining a spanning set for a vector space
Determining if vectors form a spanning set and defining a basis
Writing a vector equation using matrix multiplication
Defining multiplication of a matrix and vector and then giving an example
Introduction to the lecture on operations with matrices and vectors.
Introduction to matrices
Explaining how techniques to solve scalar equations can be used for matrix equations
Explaining the gradient vector for functions of two derivatives and its properties
Finding the directional derivative to a function of two variables at a given point and direction
Defining linear independence for a set of vectors
Determining if three vectors are linearly independent
Calculating matrix and vector multiplications with transposes
Determining if sets of vectors are spanning sets
Determining if three vectors are linearly independent
Determining if sets of 3 dimensional vectors are linearly independent
Determining if its possible for the columns of a matrix to be linearly independent
Determining if combinations of linearly independent vectors are also linearly independent
Determining whether polynomials are linearly independent
Determining whether trigonometric functions are linearly independent
Determining if two vectors are a basis for a vector space
Showing that sets of vectors are linearly independent or dependent and finding their span
Determining if three vectors form a basis for \(\mathbb{R}^3\)
Showing that a set of polynomials forms a basis for all polynomials with degree two or less
Finding the standard coordinates for a linear combination of vectors
Finding the coordinates of vectors in terms of a given basis
Finding the transition matrix from the standard basis to a given basis for a vector space
Changing the coordinates of a vector from one basis to another
Finding the coordinate vector of a quadratic given a basis for the vector space
Finding basis vectors that will make a given matrix the transition matrix for that basis
Finding the matrix representation for a linear transformation of vectors
Finding the matrix representation of a linear transformation of vectors
Writing the equation of a line passing through two points
Writing the equation of a line through two points in slope-intercept form
Writing the equation of a graphed line in standard form
Factoring a quadratic polynomial
Factoring a quadratic function
Factoring a cubic polynomial by grouping
Finding the greatest common factor of a polynomial and then factoring it out
Factoring a quadratic where the leading coefficient is 1
Factoring quadratics where the leading coefficient is not 1
Factoring several polynomials by grouping
Factoring a difference of two squares
Factoring a sum of cubes and a difference of cubes
Simplifying a rational expression
Adding and then simplifying two rational expressions
Multiplying two rational expressions and then simplifying
Dividing rational expressions and then simplifying the answer
Adding two rational expressions and then simplifying the answer
Finding any holes, vertical asymptotes, and horizontal asymptotes of a rational function
Finding any holes, vertical asymptotes, and horizontal asymptotes of a rational function
Matching a rational function with its graph
Using the quadratic formula to find the roots of a function
Finding the roots of a cubic polynomial
Finding the roots of a square root function
Finding the roots of a function with an absolute value
Finding the domain of a rational function
Finding the domain of a rational function
Finding the domain of a function with a fraction and radicals
Finding the domain of a function with a quadratic inside a square root
Finding the composition of two functions
Finding the composition of a rational function and a quadratic
Finding the composition of two functions with exponentials and logarithms
Finding the composition of two functions with a square root and a fractional exponent
Finding the composition of two functions with a cube root and a fractional exponent
Finding the inverse of a linear function and its domain
Finding the inverse of a square root function and its domain
Finding the inverse of a rational function and its domain
Finding the inverse of a rational function and its domain
Finding and simplifying the difference quotient for a quadratic function
Finding and simplifying the difference quotient for a rational function
Finding and simplifying the difference quotient for a square root function
Solving an inequality with a polynomial by factoring and using a sign diagram
Solving an inequality with a rational expression by factoring and using a sign diagram
Solving an inequality with two rational expressions using a sign diagram
Solving an inequality with a rational expression using a sign diagram
Solving an inequality with a quadratic by factoring and using a sign diagram
Finding the absolute maximum or absolute minimum of a quadratic
Writing the standard form of quadratic function given its vertex and one other point
Solving a system of two linear equations using the method of substitution
Solving a system of two linear equations using the method of substitution
Solving a system of two linear equations
Solving a system on two equations where one is linear and one is quadratic
Solving a equation with an exponential function
Solving an equation with two exponential functions with different bases
Solving an equation with exponential functions by factoring
Solving a equation with exponential functions by factoring
Finding the size of a population of rabbits with exponential growth
Finding the size of a population of rabbits with exponential decay
Finding the horizontal asymptote of an exponential function
Finding the horizontal asymptote of an exponential function
Finding the horizontal asymptote of an exponential function
Expanding a logarithmic expression using logarithm properties
Condensing logarithmic expressions into a single logarithm
Finding the exact value of logarithms using logarithmic properties
Finding the vertical asymptote of a logarithmic function
Finding the vertical asymptote of a logarithmic function
Finding the domain and x-intercepts of a logarithmic function
Finding the domain and x-intercepts of a logarithmic function
Solving a logarithmic equation using properties of logarithms
Solving a logarithmic equation using properties of logarithms
Finding the values of all six trigonometric functions of an angle given a right triangle
Evaluating the other five trigonometric functions for an acute angle given the value of sine
Evaluating the other five trigonometric functions given sine and that tangent is negative
Finding the solutions to an equation with tangent in a given interval
Solving a trigonometric equation with sine and cosine on a given interval
Solving an quadratic-like equation with sine by factoring
Solving an equation with cosine on a given interval
Solving a trigonometric equation with sine on a given interval
Sketching one period of a transformed sine graph
Sketching one period of a cosine function with transformations
Finding the amplitude, period, phase shift, and vertical shift for a given sine function
Writing a cosine function for a given graph
Introducing the idea of joint distributions for multiple random variables
First lecture in a workshop for Linear Algebra for Data Science
Review of vector functions and parametric equations
Review of limits, continuity, and the Intermediate Value Theorem
Review of the limit definition of a derivative and calculating the derivative