Question

Can we multiply two vectors?

Answer 1

Hint:Multiplication is one of the four elementary mathematical operations, the others being addition, subtraction and division. Here, you are asked to find out whether you can multiply two vectors or not. In order to answer this question, first you need to consider the definition and properties of vectors and then think if we can multiply them, if yes, then think of how we can multiply them.

Complete answer:
Vector is basically a quantity which has magnitude as well as direction. So, let us take an example of force. Let us say that you are applying a force on a block. That force of yours will actually have some direction and also a magnitude. The magnitude and direction may be changing depending on how you apply the force. So, vectors are drawn as arrows and the arrow is directed somewhere and the arrow has its own length denoting the magnitude of the vector. Now that you have an idea of what a vector is, the question arrives is whether we can multiply two vectors or not?

Let us consider two vectors. We can actually perform multiplication, but the multiplication is not just as simple as multiplying two numbers. There are two multiplications that can be performed on two vectors. One is dot product and the other is cross product. The dot product of two vectors is obtained by multiplying the magnitudes of the vectors with cosine of the angle between them and it is denoted as ${\mathbf{a}}.{\mathbf{b}}$.

Mathematically, we have ${\mathbf{a}}.{\mathbf{b}} = \left| {\mathbf{a}} \right|\left| {\mathbf{b}} \right|\cos \theta $. On the other hand, cross product is obtained by multiplying the magnitudes of the vectors with sine of the angle between them and also a normal vector $\widehat n$ which is perpendicular to the plane containing the two vectors. It is denoted as \[{\mathbf{a}} \times {\mathbf{b}}\] and is given as \[{\mathbf{a}} \times {\mathbf{b}} = \left| {\mathbf{a}} \right|\left| {\mathbf{b}} \right|\sin \theta \widehat n\].

So, yes, we can multiply two vectors.

Note:You are supposed to remember that the multiplications in vectors are of two types. One is dot and the other is cross. Dot product is also called scalar product because the product is a scalar quantity and the cross product is also called vector product because the product is a vector quantity. Keep in mind the mathematical form of each product.