Since this product has magnitude and direction, it is also known as the vector product. The symbol used to represent this operation is a large diagonal cross (×), which is where the name "cross product" comes from. Geometrically, the cross product of two vectors is the area of the parallelogram between them. A = AA cos 0° = A xA x + A yA y + A zA zĪ 2 = A x 2 + A y 2 + A z 2 cross product.From this we can derive the Pythagorean Theorem in three dimensions. The dot product of two vectors is thus the sum of the products of their parallel components. The resulting product looks like it's going to be a terrible mess, but consists mostly of terms equal to zero. Using this knowledge we can derive a formula for the dot product of any two vectors in rectangular form. In addition, since a vector has no projection perpendicular to itself, the dot product of any unit vector with any other is zero. Since the projection of a vector on to itself leaves its magnitude unchanged, the dot product of any vector with itself is the square of that vector's magnitude.Īpplying this corollary to the unit vectors means that the dot product of any unit vector with itself is one. Since this product has magnitude only, it is also known as the scalar product.
), which is where the name "dot product" comes from.The symbol used to represent this operation is a small dot at middle height ( Geometrically, the dot product of two vectors is the magnitude of one times the projection of the second onto the first. Multiplication of a vector by a scalar is distributive.Ĭonsequently, the rectangular form vector… The scalar changes the size of the vector. Multiplication of a vector by a scalar changes the magnitude of the vector, but leaves its direction unchanged.