Sphere,
Distance in 3D,
Triangle determined by 3 points,
Collinear Points,
Coplanar points and vectors,
Vector through 2 Points,
Vector Sum,
Vector Difference,
Unit Vector,
Dot Product,
Vector Cross Product,
Scalar Triple Product,
Equations of Lines in 3D,
Equations of Planes in 3D
Vector Functions
3 Dimensions and Vectors
3D Coordinate Systems
Sphere
From Quadratic Equation
From Center and Radius
Distance in 3D
First Octant
Not First Octant
Distance from a Point to a Plane
Triangle determined by 3 points
Equilateral
Isosceles
Right
Neither
Collinear
Three Points
Not Collinear
Rectangular Box
Vectors
Vector through 2 points
Vector In 2D P(2,-5); Q(-3,1)
Vector In 3D
Vector Sum
Vector Sum In 2D
Vector Sum In 3D
Vector Difference
Vector Difference In 2D
Vector Difference In 3D
Difference of Scalar Multiples
Graphical Addition and Subtraction in 2D
Add Vectors
Subtract Vectors
Add 3 Vectors
Add 3 Vectors to 0
Add Parallel Vectors
Subtract Parallel Vectors
Add Multiples of Parallel Vectors
Subtract Multiples of Parallel Vectors
Subtract Multiples of Parallel Vectors to Zero Vector
Unit Vector Parallel to a Given Vector
Unit Vector in 2D
Unit Vector In 3D
Simultaneous Vector Equations
Dot Product
2D Dot Product
3D Dot Product
where
, and
.
Dot Product using magnitudes and angle
Angle between 2 Vectors in 2D
Angle between 2 Vectors in 3D
where
, and
.
Triangle: 3 points: Find 3 angles
A(2, 2, -2), B(-4, 4, 2), C(2, -5, 4).
Skew Vectors in 2D
Skew Vectors in 3D
Skew Vectors in 3D
where
and
.
Parallel Vectors in 2D
Parallel Vectors in 3D
where
and
Orthogonal Vectors in 2D
Orthogonal Vectors in 3D
where
and
Unknown Components of Orthogonal Vectors in 2D
Unknown Components of Orthogonal Vectors in 3D
Scalar and Vector Projections in 2D
Scalar and Vector Projections in 3D
Vector Cross Product
Via Determinant, 2D
Via Determinant, 3D
Via Magnitudes and Angle
Unit Vectors Orthogonal to 2 vectors
Vector Orthogonal to a Plane
Area of Parallelogram given by A(6, – 3, – 2), B(3, – 3,2), C( – 4,8,4), and D( – 1,8,0)
Area of Triangle from 3 Points
Scalar Triple Product
Volume of Parallelopiped, adjacent edges AB, AC, AD
3 Coplanar Vectors
4 Coplanar Points
Equations of Lines in 3D
Find the vector, parametric, and symmetric equations for the line L through P(2,-3,2)
and parallel to
.
Find the vector, parametric, and symmetric equations for the line L through A(7,–2, 5) and B(–5,–6,5).
Show that the line L1 through the points A(7, –2, 3) and B(–5, –6, 5)
is parallel to the line L2 through C(3, 5, 1) and D(–3, 3, 2).
Show that the line L1 through the points A(7, –2, 3) and B(–5, –6, 5)
is perpendicular to the line L2 through C(–1, 1, 2) and D(–3, 6, 0).
Find the parametric, symmetric, and vector equations for the line L1 through A(9, –2, 3) and parallel to the line L2 having the parametric equations
Find the symmetric and parametric equations of the line through point A(4,7,5) perpendicular to the plane
Show that two lines are parallel
L1:
,
L2:
Show that two lines intersect
L1:
,
L2:
Show that two lines are skew
L1:
,
L2:
Equations of Planes in 3D
Plane through a Point with a given Normal Vector: Example 1
Plane through a Point with a given Normal Vector: Example 2
Plane through a Point Parallel to another Plane
Plane through 3 Points: Example 1
A(1, -6, 0), B(-4, 2, -5), C(-2, 4, 1): solution
Plane through 3 Points: Example 2
A(1,0,2), B(4,0,2), and C(2,3,0): details
Plane through 3 Points: Example 3
A(-2, -2, 3), B(2, -1, 2), and C(3, 6, 2): details
Plane through a Point, Containing a Given Line: Example 1
Plane through a Point, Containing a Given Line: Example 2
Intersection of a Line and a Plane
Direction Numbers for the Line of Intersection of 2 Planes
Cosine of the Angle between 2 Planes
Determine the Angle between 2 Planes
Line of Intersection of 2 Planes: Example 1
Line of Intersection of 2 Planes: Example 2
Vector Functions
Find the limit
Find the limit
Find the derivative
Find the domain
Plane Curves
Sketch Curve; Position & Tangent Vectors
Sketch Curve; Position & Tangent Vectors
Sketch Curve; Position & Tangent Vectors
Position & Tangent Vectors
Unit Tangent Vector of Helix
Arc Length
Arc Length for a Helix
Curvature
Curvature of a 3D Parabola
Curvature of a 3D Trig Curve
Curvature of a 3D Twisted Cubic
