Outline of Vector Items

Vectors/Basic Facts

Misconceptions or Tabula Rasa. Students are somewhat afraid of vectors, sense that there is something different about them, but don't quite know what it is. They are familiar with numbers and don't understand that vectors are different algebraic objects for which there are certain rules of composition with which these objects can be handled safely. They don't have a concept of different kinds of algebraic objects subject to specific rules of composition.

Students want to treat vectors as numbers. They are reluctant to use a different notation (with arrow) to denote vectors. Those who have seen vectors before in high school want to describe vectors in terms of magnitude and direction, not in terms of components. They usually have never heard of components. When asked to calculate a vector quantity, e.g., a force, students tend to want to calculate the magnitude only. They think they are not finished if they have found the components of the force.

Some students have difficulty with the graphical representation by vectors in terms of arrows. They don't understand in which direction the arrow points, have trouble identifying the tail and the tip of an arrow. They don't understand that the length of the arrow represents the magnitude of the vector. They don't understand the concept of magnitude of a vector.

Some students think that vectors need to be shown against axes. They don't have the notion of a vector as an absolute geometric entity. (It is true that in order to describe the direction of a vector one needs a reference line. Still the notion of axes is something separate. The notion of axes is related to the notion of basis of a vector space, which is not essential in defining the vector concept in terms of a set of objects with certain algebraic properties.)

Some students don't understand clearly that vectors can be dragged around and don't change in the process.

There are specific misconceptions related to the algebraic rules governing vectors which will be listed under Vectors/Addition, etc.

Vectors/Scalar Multiplication.

This is not too difficult. Students need to learn about the fact that there is such a thing as scalar multiplication, what it does, especially the multiplication by a negative scalar. They need to learn what the negative of a vector is.

Vectors/Addition

Misconceptions or Tabula Rasa. In adding two vectors, students ignore vector nature and add just magnitudes. Some students realize that vectors are different, but just don't know how to add them and have no intutitive feeling for what the resultant might be. Or, if they know about the tip-to-tail, they add the vectors tip-to-tail and then draw the resultant in the opposite direction, from the free tip to the free tail, so that the three vectors form a closed circuit. Or, they don't realize that there is a resultant: adding two vectors means joining them, but not drawing the resultant. Students are not familiar with the parallelogram construction. In adding more than two vectors, students are lost, don't have the understanding that they can use the associative law, first add two and then the third one to the resultant of the two.

Vectors/Subtraction

Misconceptions or Tabula Rasa.