Scalar Product of Vectors

The scalar product and the vector product are the two ways of multiplying vectors which see the most application in physics and astronomy. The scalar product of two vectors can be constructed by taking the component of one vector in the direction of the other and multiplying it times the magnitude of the other vector. This can be expressed in the form:

If the vectors are expressed in terms of unit vectors i, j, and k along the x, y, and z directions, the scalar product can also be expressed in the form:

Scalar Product Calculation

You may enter values in any of the boxes below. Then click on the symbol for either the scalar product or the angle. The vectors A and B cannot be unambiguously calculated from the scalar product and the angle. If the angle is changed, then B will be placed along the x-axis and A in the xy plane.

The scalar product = ( )( )(cos ) degrees.

()()+()()+()() =

Note: The numbers above will not be forced to ba consistent until you click on either the scalar product or the angle in the active formula above.

Scalar Product Applications

Geometrically, the scalar product is useful for finding the direction between arbitrary vectors in space. Since the two expressions for the product:

involve the components of the two vectors and since the magnitudes A and B can be calculated from the components using:

One important physical application of the scalar product is the calculation of work:

The scalar product is used for the expression of magnetic potential energy and the potential of an electric dipole.