Analytical Method for Addition and Subtraction of Vectors
Although graphical way is nice to visualize vectors in two dimensions, it becomes difficult to work with it in three dimensions, and also when many vectors and many operations with them are involved. So vector algebra is best done by representing them in terms of their components along the x, y & z axes in space. We now discuss how to this is done.
To represent vectors in terms of their x,y and z components, let us first introduce the concept of unit vector. A unit vector in a particular direction is a vector of magnitude ‘1’ in that direction. So a vector in that particular direction can be written as a number times the unit vector . Let us denote the unit vector in x-direction as , in y-direction as and in z-direction as . Now any vector can be described as a sum of three vectors , and in the directions x, y and z, respectively, in any order (recall that order does not matter because vector sum is commutative). Then a vector
It is clear from figure 9 that the magnitude of the vector is . Now when we add two vector, say and , all we have to do is to add their x-components, y-components and the z-components and then combine them to get
Similarly multiplying a vector by a number is same as increasing all its components by the same amount. Thus
How about the multiplying by -1? It just changes the sign of all the components. Putting it all together we see that
Having done the addition and subtraction of two vectors, we now want to look at the product of two vectors. Let us see what all possible products do we get when we multiply components of two vectors. By multiplying all components with one another, we have in all nine numbers shown below:
The question is how do we define the product of two vectors from the nine different numbers obtained above? We will delay the answer for some time and come back to this question after we establish the transformation properties of scalars and vectors. By transformation properties we mean how does a scalar quantity or the components of a vector quantity change when we look at them from a different (rotated) frame?
Let us first look at a scalar quantity. As an example, we take the distance traveled by a person. If we say that the distance covered by a person in going from one place to another is 1000m in one frame, it remains the same irrespective of whether we look at it from the frame (xy ) or in a frame (x’ y‘ ) rotates about the z-axis (see figure 10).
Let us now say that a person moves 800 meter along the x-axis and 600 meters along the y-axis so that his net displacement is a vector of 1000m in magnitude at an angle of from the x-axis as shown in figure 10. The total distance traveled by the person is 1400m. Now let us look at the same situation frame different frame which has its x’ & y’ axis rotated about the z- axis. Note that the total distance traveled by the person (a scalar quantity) remains the same, 1400m, in both the frames. Further, whereas the magnitude of the displacement & its direction in space remains unchanged, its components along the x’ and y’ axis, shown by dashed lines in figure 10, are now different. Thus we conclude the scalar quantity remains unchanged when seen from a rotational frame. The component of a given vector are however different in the rotated frame, as demonstrated by the example above. Let us now see how the components in the original frame and the rotated frame are related.
In figure 11, OA is a vector with Ax = OB , Ay = AB, Ax’ = OA’ and Ay’ = AA’. Using the dashed lines drawn in the figure, we obtain
So we learn that if the same vector is observed from a frame obtained by a rotation about the z-axis by an angle θ, its x and y components in the new frame are
One can similarly define how components mix when rotation is about the y or the x- axis. Under the y axis rotation
And under a rotation about the x-axis
Let us summarize the results obtained above:
- Scalar quantity is specified by a number and that number remains the same in two different frames rotated with respect to each other.
- A vector quantity is specified by its components along the x, y, and the z axes and when seen from another frame rotated with respect to a given frame, these components change according to the rules derived above.