Chapter 6, Problem 6PS

To determine

To find: Write the vector $w$ in terms $u$ and $v$ .

Answer to Problem 6PS

  $w$ in terms of $u$ and $v$ is $w=\frac{v+u}{2}$

Explanation of Solution

Given information:

Terminal point of $w$ bisects the line segment.

The line segment joining the terminal points of the vector $u,v$ and $w$ is bisected by the vector $w$ and thus divided into two vectors $x$ and $y$ of same magnitude but opposite directions as can be seen from the figure.

Hence the relation

  $x=-y$ --------(1)

The relation between the given vectors can be determined if we consider the triangle formed by them separately.

Since the vector $v$ is the resultant of the vector $w$ and $x$ .Vectorially, this can be written as follows:

  $v=w+x$ -------(2)

Substituting value of $x$ from equation-(1) in equation-(2)

We get

  $\begin{array}{l}v=w+x\\ \Rightarrow v=w-y\end{array}$

  $v=w-y$ --------(3)

Similarly the vector $u$ is the resultant of the vector $w$ and $y$ .Vectorially ,this can be written as follows:

  $u=w+y$ ------(4)

Adding equation-(3) and equation-(4)

We get

  $\begin{array}{l}v=w-y\\ u=w+y\\ \Rightarrow v+u=w-y+w+y\\ \Rightarrow v+u=w+w-y+y\\ \Rightarrow v+u=2w\\ \Rightarrow 2w=v+u\\ \Rightarrow w=\frac{v+u}{2}\end{array}$

Therefore,

  $w$ in terms of $u$ and $v$ is $w=\frac{v+u}{2}$