Vectors and Interactive Applets

A vector is a quantity that has both magnitude and direction, in contrast to a scalar, which has magnitude alone. Vectors model physical quantities such as displacement, velocity, acceleration and force, and they are usually drawn as arrows whose length represents magnitude and whose orientation represents direction. In a coordinate system a vector can be described by its components along each axis, which makes algebraic manipulation straightforward while preserving the geometric picture.

The basic operations on vectors all have clear geometric meanings that applets can animate. Adding two vectors corresponds to placing them head to tail, with the sum running from the start of the first to the end of the second; the same result follows from the parallelogram rule. Subtraction reverses one vector before adding, and multiplying by a scalar stretches, shrinks or flips a vector without changing the line it lies along. Dragging the arrows in an applet and watching the resultant update makes these rules immediate.

Two products combine vectors in different ways. The dot (scalar) product multiplies their magnitudes by the cosine of the angle between them, yielding a number that is largest when the vectors are parallel and zero when they are perpendicular; it is used to find angles and projections. The cross (vector) product, defined in three dimensions, produces a new vector perpendicular to both inputs, with a magnitude equal to the area of the parallelogram they span. These operations underpin much of physics and engineering, from computing work done by a force to describing rotation and torque.

Frequently asked questions

What is the difference between a vector and a scalar?

A scalar has only a size, such as temperature or mass. A vector has both a size and a direction, such as velocity or force.

How are two vectors added?

They are placed head to tail and the sum is the arrow from the start of the first to the end of the second, equivalent to adding their components.

What does the dot product tell you?

It gives a number related to how aligned two vectors are; it is zero when they are perpendicular and is used to find angles and projections.