Vector Geometry

IB Higher Level Mathematics : Core

4. Vector Geometry

4.1

Vectors as displacements in the plane and in three dimensions, v =

For an introduction to the subject that aims at a "visceral understanding rather than a rigorous logical presentation", see Vectors. Explorations are conducted using an interactive program, The Geometer's Sketchpad. A demo version of the program can be downloaded from the site (Mac or Windows). The site also has a Vector Discussion Group attached to it.

The Graphing Vector Calculator is an interactive Java applet that enables the graphing of two-dimensional vectors and the results of simple operations on them.

Components of a vector; column representation.

See Decomposition of Vectors, from the Vectors site.

The vector 8i + 15j + 9k, graphed using the TI-83 calculator. An article on the graphing of vectors appeared in the Winter 1998 edition of Eightysomething!, the newsletter for users of TI graphing calculators. A program to calculate the vector product is also available.

The sum of two vectors; the zero vector; the inverse vector, -v.

Mutiplication by a scalar, kv.

The vector sum i + j + k.
Graphed using Mathematica and METRIC packages.

Magnitude of a vector, |v|.

Position vectors

= a.

Unit vectors including i, j and k.

See Vector Length and Unit Vectors, both from the Vectors site.

4.2

The scalar product of two vectors:

Properties of the scalar product

Perpendicular vectors; parallel vectors.

See Dot Products and Projections, from the Oregon State University Math Department's Web Study Guide.

The component formula for the scalar product, together with examples and problems, can be found at The Scalar Product, from S.O.S. Mathematics.

Also see Definition of the Dot Product, What Good Are Dot Products?, and Projection - Theory, all from the Vectors site from The Math Forum.

The projection of one vector onto another.
From Projection - Numbers. Used with permission.

4.3

The expression v

w= |v||w| cos q; the angle between two vectors.

The projection of a vector v in the direction of w; simple application, e.g. finding the distance of a point from a line.

4.4

The vector product of two vectors |v x w| = |v||w| sin q.

The formula for the area of a triangle in the form ½ |v x w|

See The Vector Cross Product - A JAVA Interactive Tutorial, from the University of Syracuse. An interactive diagram of a × b = c allows vectors a and b to be lengthened and shortened, and the angle f between them can be adjusted. The result of these changes can be seen in vector c. The plane containing vectors a and b can also be tilted. Recommended.

The Cross Product provides a good illustration of the vector product, making clear the length of the vector formed when two vectors are crossed.

Also see The Cross Product, from the OSU Math Department's Web Study Guide.

The vector product i × j.
Vector i is red. The cross product is shown in magenta, perpendicular to the xy-plane.
The diagram was made using Mathematica and METRIC packages.

4.5

Vector equation of a line r = a + lb.

Vector equation of a plane r = a + lb + mc

Use of normal vector to obtain r

n = a

n.

Cartesian equations of a line and plane.

See Equations of Lines and Planes, from the Oregon State University Math Department's Web Study Guide.

Also see 12.2 Planes, from Geometry Formulas and Facts. This resource is from the Geometry Center at the University of Minnesota.

The Three Dimensional Graphing Applet can be used to graph a plane expressed in Cartesian form. The plane can be rotated, and the viewer can zoom in and out. The applet was written by James Goodenberger, a fifteen year old high school student.

4.6

Intersections of
(i) two lines;
(ii) a line with a plane;
(iii) two planes;
(iv) three planes.

The intersection of three planes in space.
The red plane has equation z = 3. The equation of the green plane is -4y - 3z = 8. The equation of the cyan plane is 4x + y + 2z = 2. The planes meet at the point (-1.75, 3, 3). Created using Mathematica and packages developed by METRIC.

4.7

Distances in three dimensions between points, lines and planes.