| Pre-calculus
Contents - C |
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Coordinate
Geometry or
Analytic Geometry - Conic Sections |
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Circle |
General equation of a circle with the center
S(p, q) - translated circle |
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The
equation of the circle, example |
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Equation of the circle with the
center at the origin O(0, 0)
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Circle through three points
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The
equation of the circle through three points, example |
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Circle
and Line
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Line circle intersection
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Circle and line,
examples
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Equation of a tangent at a point
of a circle with the center at the origin
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Equation of a tangent at a point
of a translated circle
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Equation of a tangent at a point
of a circle, examples
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Condition of tangency - Condition for a line to be the tangent to a circle
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Condition for a line to be the tangent to the circle with
the center at the
origin
O(0, 0)
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Condition for a line to be the tangent to the translated circle
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Tangents to a circle from a point outside the circle - use of the
tangency condition
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Angle between a line and a circle
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Mutual position of two circles
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The radical line or the radical
axis
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The pole and the polar
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Angle between two circles
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Ellipse
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Definition and construction of an ellipse |
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Eccentricity and linear
eccentricity
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Constructions of an ellipse |
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Equation of the ellipse, standard
equation of the ellipse
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Major axis, minor axis, and
vertices
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The focal parameter, latus rectum
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The parametric equations of the ellipse |
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Equation of a translated ellipse
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Ellipse
and Line |
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Intersection of ellipse and line - tangency condition
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Equation of the tangent at a point on the ellipse
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Construction of the tangent at a point on the ellipse
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Angle between the focal radii at a point of the ellipse
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Tangents to an ellipse from a
point outside the ellipse - use of the tangency condition
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Construction of tangents from a point outside the ellipse
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Polar and pole of the ellipse
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Equation of the polar of the given
point
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Ellipse and line examples
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Hyperbola
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Definition and construction
of the hyperbola
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Construction of the hyperbola
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Equation of the hyperbola
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Properties of the hyperbola
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Examining equation of the hyperbola
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Equilateral or rectangular
hyperbola
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Translated hyperbola
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Equation of the hyperbola in vertex form
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Parametric equations of the hyperbola
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Examples of hyperbola |
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Equilateral or rectangular hyperbola with the coordinate axes as its
asymptote
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Translation
of equilateral or rectangular hyperbola with the coordinate axes as its
asymptote |
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Hyperbola
and line
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Hyperbola and line relationships
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Condition for a line to be the tangent to the hyperbola -
tangency condition |
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The equation of the tangent at the point on the hyperbola
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Polar and pole of the hyperbola
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Construction of the tangent at the point on the hyperbola
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Construction of tangents from a point outside the hyperbola
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Properties of the hyperbola
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The area of a triangle which the tangent at a point on the hyperbola forms with
asymptotes |
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The tangency point bisects the line
segment of the tangent between
asymptotes |
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The parallels to the asymptotes through the tangency point intersect asymptotes |
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The equation of the equilateral or rectangular hyperbola with the
coordinate axes as its
asymptotes |
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Hyperbola and line examples
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Parabola
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Definition and construction
of the parabola
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Construction of the parabola
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Vertex form of the equation of a parabola
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Transformation of the equation of a parabola
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Equation of a translated parabola - the standard form
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| The parabola whose axis
of symmetry is parallel to the y-axis |
| Equations of the parabola
written in the general form |
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Parametric equations of the parabola
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Examples
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Parabola and line |
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Common points of a line and a parabola |
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Condition for a line to be the tangent to the parabola
- tangency condition
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The equation of the tangent and the normal at the point on the parabola
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Properties of the parabola
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Polar of the parabola
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Construction of the tangent at the point on the parabola
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Construction of the tangents from a point exterior to the parabola
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Parabola and line, examples |
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Conics, a Family of Similarly Shaped Curves
- Properties of Conics |
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Conics,
a Family of Similarly Shaped Curves - Properties of Conics |
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Dandelin's
Spheres - proof of conic sections focal properties
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Proof that conic section curve is
the ellipse
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Proof that conic section curve is the hyperbola
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Proof that conic section curve is the parabola
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Conics - a family of similarly shaped curves |
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Vectors in a Plane and
Space |
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Vectors
in a Plane |
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Vectors - introduction |
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Length, magnitude or
norm of the vector |
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Collinear, opposite
and coplanar vectors |
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Addition of vectors |
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Triangle rule (law)
and parallelogram rule |
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Zero or null vector |
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Subtraction of vectors |
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Scalar multiplication
or multiplication of a vector by scalar |
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Unit vector |
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Addition, subtraction
and scalar multiplication of vectors, examples |
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Linear combination of
vectors |
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Linear dependence of
vectors |
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Vectors
and a coordinate system, Cartesian vectors |
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Vectors in
a coordinate plane (a two-dimensional system of coordinates), Cartesian
vectors |
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Radius vector or position vector |
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Vector components |
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Vectors in
a two-dimensional system, examples |
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Vectors in
three-dimensional space in terms of Cartesian coordinates |
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Angles of
vectors in relation to coordinate axes, directional cosines - scalar
components of a vector |
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The unit vector of a vector |
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Vectors in
a three-dimensional coordinate system, examples |
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Scalar product or dot
product or inner product |
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Orthogonality or
perpendicularity of two vectors |
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Different positions of two vectors and the corresponding values of
the scalar product |
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Square of magnitude of
a vector |
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Scalar product of unit
vectors |
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Scalar or dot
product properties |
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Scalar product in the
coordinate system |
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Angle between vectors
in a coordinate plane |
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Projection of a vector in the direction of another vector,
the scalar and vector components |
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Vector
product or cross product |
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Vector
product |
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Right-handed system |
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An example for the vector product in physics |
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The condition for two vectors to be parallel |
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The vector products of the standard unit vectors |
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The vector product properties |
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The vector product in
the component form |
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The mixed product
or the scalar triple product |
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The mixed product
or scalar triple product definition |
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The
mixed product properties |
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The condition for
three vectors to be coplanar |
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The mixed product
or scalar triple product expressed in terms of components |
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The vector product
and the mixed product use, examples |
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| Points,
Lines and Planes in Three-dimensional Coordinate System Represented by
Vectors |
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Line in a
three-dimensional (3D) coordinate system |
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Equations of a line in space |
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The
vector equation of a line
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The parametric equations of a line
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Equation of a line defined by
direction vector and a point - Symmetric equation of a line |
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Line given by two points
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Distance between two given points
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Orthogonal projection of a line in
space onto the xy
coordinate plane |
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A
line in a 3D space examples |
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Angle between lines
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Condition for intersection of two
lines in a 3D space
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Plane in a three-dimensional (3D) coordinate system |
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Equations of
a plane in a coordinate space
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The equation of a
plane in a 3D coordinate system |
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The Hessian normal form of the
equation of a plane
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The intercept
form of the equation of a plane |
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A plane in a 3D
coordinate system, examples |
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Equation of the
plane through three points, example |
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The distance between a point and a
plane - plane given in Hessian normal form
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Comparison of
the general form and the
Hessian normal form of equations of a plane
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The distance of a point to a plane -
plane given in general form
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Angle (dihedral angle) between two
planes
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Line and Plane
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The line of intersection of two planes |
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Projection of a line onto coordinate planes
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How determine two planes of which, a given line is their
intersection line
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Intersection point of a line and a plane
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Sheaf or
pencil of planes
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Points, Lines and planes relations in 3D space, examples
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The angle between line and plane
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Point,
Line and Plane - orthogonal
projections, distances, perpendicularity of line and plane
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Through a given point pass a line perpendicular to a given plane
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Given a line and a point, through
the point lay a plane perpendicular to the line
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Projection of a point onto a plane
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Projection of a point onto a line
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Through a given point
pass a line perpendicular to a given line,
example
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Projection of a line onto a plane,
example
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Plane laid through a given point, such that be parallel with two
skew lines, example |
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Plane laid through a given point, such that be parallel with two
parallel lines, example |
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Distance between
point and line, example |
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Distance between point and plane,
example |
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Distance between parallel lines,
example |
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Distance between two
skew lines, example |
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Matrices
and Determinants |
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Matrix
definition |
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Square matrices |
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The identity matrix |
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Vectors |
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The transpose of a
matrix |
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Basic
matrix operations |
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Matrix addition and
subtraction |
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Scalar
multiplication of a matrix |
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Matrix multiplication |
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The
matrix-vector product
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The determinant of a
matrix |
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Expanding a determinant
by cofactors
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Calculating the value
of a determinant |
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The
determinant of a 2
´
2
matrix |
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The
determinant of a 3
´
3
matrix |
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Properties of determinants |
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The inverse of
a matrix |
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Systems of Linear
Equations |
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Solving systems of
equations using matrices |
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Inverse matrix method |
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Cramer’s rule |
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Cramer’s Rule and
inverse
matrix method correlation |
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Cramer’s
rule (using the determinant) to solve systems of linear equations
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Solving system
of two equations in two unknowns using Cramer's rule
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Solving system of three equations
in three unknowns using Cramer's rule
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Method
of expanding a determinant of a rank n by cofactors |
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Sequences
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Arithmetic sequence/progression
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General term of an arithmetic
sequence
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The sum of the first n terms of an arithmetic sequence |
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The sum of the first n natural numbers |
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Arithmetic sequences, examples
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Geometric sequence/progression |
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General term of a geometric sequence
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The sum of the first n terms of a finite geometric
sequence, geometric series |
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Geometric sequences, examples
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Recursive
definition and the recursion
formula |
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Series
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Finite and infinite, convergent and
divergent series |
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Geometric series
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The sum of an infinite geometric
sequence, infinite geometric series |
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The sum of an infinite converging
geometric series, examples
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Converting recurring decimals
(infinite decimals) to fraction
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Permutations,
Combinations and Variations
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Permutations
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Permutations
of n objects some of which are the
same |
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Combinations
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Combinations
with repetition
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Variations |
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Variations or permuted
combinations (permutations without repetition) |
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Variations with repetition (or permuted
combinations with repetition) |
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Mathematical
Induction
and The Binomial Theorem
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The principle of
mathematical induction |
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Proving a statement, a
theorem or a formula |
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Mathematical
induction examples
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The Binomial Theorem
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Factorial
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Binomial coefficients and
Pascal's triangle |
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The
binomial theorem, sigma notation
and binomial expansion
algorithm |
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The
binomial theorem and binomial expansion algorithm examples |
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| Back
to pre-calculus contents - B |
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