Question

What does a dot product equal to one mean?

Answer 1

Hint: We need to have a basic idea of what a dot product actually means. There are many definitions of dot products. We use the basic definition which states dot product or scalar product is given by: $$\left|A\right|\cdot \left|B\right|=\left|A\right|\cdot \left|B\right|\cos \theta$$ where $$\theta$$ is the angle between two given vectors.


Complete step by step solution:
The dot product is also called a scalar product. This states that multiplying any two vectors will give us a single vector which is scalar. Here we have considered the vectors as A and B
Scalar product or dot product is basically defined as a product of 2 vectors along with the angle of cosine between them. Look at the formula above , we have written 2 terms $$\left|A\right|and\left|B\right|$$, these are called magnitudes of vectors(it is the length of the vector).
According to the question they have given that the dot product is 1. It means that $$A\cdot B=1$$
Or, $$\left|A\right|\cdot \left|B\right|\cos \theta =1$$
This is possible only when : $$\cos \theta =0$$,
And $$\cos \theta =0$$ , means that the angle between the 2 vectors A and B is zero.
Hence the 2 vectors are parallel. To get better idea look at the image shown below:

Now the fact that the angle between 2 vectors is 0 means that the vector could be parallel or antiparallel as shown in the image below;

Hence if the dot product is 1 it means that the two given vectors are parallel. And if the dot product is -1 it means that the two given vectors are antiparallel.

Additional Information:
$$\bullet$$ If the angle between 2 given vectors is $${90}^{\circ }$$it means that $$A\cdot B=0$$ and they are said to be orthogonal.
$$\bullet$$ he dot product of a unit vector with itself is given by: $$i\cdot i=j\cdot j=k\cdot k=1$$


Note:
We should remember that the angle being 0 doesn’t mean that the vectors are only parallel; they can also be antiparallel. Thus when the dot product is 1 it means the vectors are parallel and when the dot product is-1 it means the vectors are anti-parallel.