Is force a scalar or vector
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Scalar or inner product of vectors
Numbers can be added and multiplied. For vectors, which are actually represented by arrows, there is the clear parallelogram rule of addition. But what about the multiplication of two vectors? Is there a reasonable reason for this? And if so, how do you multiply arrows? The answer is yes and of course it is combined with mathematical definitions! There are even three of them, namely the scalar, vector and dyadic product. This chapter is dedicated to the first answer, briefly referred to as the scalar product. The second answer is another chapter on the subject of vectors. The dyadic product is treated in the matrix calculation.
For a clear introduction, let's imagine a carriage that stands on a rail. If we want to move it a short distance, we have to use a force. The work done (a scalar!) Is, according to the laws of physics, distance times force in the direction of the path. Path and force are known to be described by vectors, here symbolized with or. So we now have to think about how the product way times force is to be designed with or, so that the value of the work arises. It must be taken into account that the tensile force that is acting does not necessarily point along the rails. A person who pulls the car with a rope can walk between the rails in the gravel, but it is more convenient to walk next to the rails on a smooth path parallel to the rails. The tensile force transmitted via the tensioned pull rope therefore generally forms an angle α with the direction of travel. The following figure shows the basics of the situation.
Parameters of any vector pair and are the amounts and as well as the angle enclosed by both vectors. The wagon example makes it clear that the multiplicative combination of the amounts with the cosine of α can be useful. The following definition is therefore generally made.
- Dot product
- Let and be two vectors and the angle they include. The scalar product or inner product (read: u times v) is then defined according to
It is permissible to omit the point between the two vectors in the scalar product, i.e. to write it only briefly.
To the angle - component of one vector in the direction of the second
The restriction given in the definition of the scalar product for the angle between the two vectors is common, but actually not necessary because of the symmetry of the cosine. For this area, the example of the car immediately shows why this is so. For these angles, the pull rope simply points to the left side of the tracks, and not to the right as shown above. The resulting work is the same in both cases.
The scalar product assumes negative values for the angular range. In the wagon example, there is a negative work, which requires a comment. Before that, however, we consider a consequence of the definition of the scalar product. For the vector component of the force in the direction of the tracks we get by combining the above formula
The equation represents the force component as a multiple of the unit vector of the path. The sign of this scalar prefactor is determined by the sign of the scalar product.
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