Vectors
3.0 Addition of Vectors
3.0 Addition of Vectors
1. Triangle Law of Vector Addition
A vector represents the displacement from one point to another point. Let vector $\overrightarrow {PQ} $ represents the displacement from point $P$ to point $Q$. Now consider a situation in which a boy moves from $P$ to $Q$ and then from $Q$ to $R$ as shown in figure. The net displacement made by the boy from point $P$ to the point $R$, is given by the vector $\overrightarrow {PR} $ and represented as $\overrightarrow {PR} = \overrightarrow {PQ} + \overrightarrow {QR} $. This is called as triangle law of vector addition. The basic concept of addition of two vectors is that coincide initial point of one vector with the terminal point of other vector.
Consider two vectors $\overrightarrow {p} $ and $\overrightarrow q $. We have shifted vector $\overrightarrow q $ without changing its magnitude and direction, so that it’s initial point coincides with the terminal point of $p$. Then, the vector $\overrightarrow p + \overrightarrow q $ represented by the third side $PR$ of the triangle $ABC$, gives us the sum (or resultant) of the vectors $\overrightarrow p$ and $\overrightarrow q $ i.e. in triangle $PQR$ as shown in figure, we have $\overrightarrow {PR} $ and represented as $$\overrightarrow {PR} = \overrightarrow {PQ} + \overrightarrow {QR} $$ And we know that $\overrightarrow {PR} = - \overrightarrow {RP} $, so by using above equation, we can say that $$\overrightarrow {PQ} + \overrightarrow {QR} + \overrightarrow {RP} = \overrightarrow {PP} = \overrightarrow 0 $$
Note:
a) If the sides of a triangle are taken in an order, then it leads to zero resultant because initial and final (or terminal points) get coincide.
Now, construct a vector $QR'$ so that its magnitude is same as the vector $QR$ but direction is opposite i.e.,$$\overrightarrow {QR'} = \overrightarrow { - QR} $$
Now, applying the triangle law of vector addition, we can write as
$$\overrightarrow {PR'} = \overrightarrow {PQ} + \overrightarrow {QR'} $$$$\overrightarrow {PR'} = \overrightarrow {PQ} + \overrightarrow {(- QR')} $$$$\overrightarrow {PR'} = \overrightarrow {p} - \overrightarrow {q} $$ which represents the difference between two vectors.
b) Concept of relative velocity (studied in physics) is based on the difference of two vectors.
