To multiply a vector by a scalar quantity, multiply the magnitude of the vector by the absolute value of the scalar. The sign of the scalar dictates direction: if the scalar is positive, the resultant vector is in the same direction as the original vector; if the sign of the scalar is negative, the resultant direction is opposite to that of the original.
Multiplying two vectors uses operations known as the cross product and dot product. Graphically, two vectors being multiplied have the same origin for the tails of the vectors. The cross product is denoted as axb and yields a vector. The magnitude of the resultant vector c is given by
(4) c = absin f, where f is the angle between the vectors.
The direction of c is always perpendicular to the plane containing the vectors a and b. The right hand rule is used to determine in which direction the vector actually points. When the right hand is aligned with the first vector (in this case a because axb is being considered) such that the fingers curl toward the second vector b (in the direction of the smaller angle), the thumb points in the direction of the resultant c.
In contrast, the result of a dot product a•b of two vectors yields a scalar quantity. The dot product is calculated by
