Question

26 N force is applied by direction of \[5\overset{\hat{\ }}{\mathop{i}}\,-12\overset{\hat{\ }}{\mathop{j}}\,\]. Force \[\overset{\to }{\mathop{F}}\,\]=?

Answer 1

Hint: The vector force is the product of the magnitude of the force and the unit vector along with that force. A unit vector is the ratio of the given direction vector by its magnitude. The magnitude is computed as the root of the squares of the coefficients.

Formula used: \[\overset{\to }{\mathop{F}}\,=\,|F|\overset{\hat{\ }}{\mathop{n}}\,\]

Complete step by step answer:
From given, we have the data,
The magnitude of the force = 26 N
\[|F|=26\,N\]
The direction of the force in the vector form = \[5\overset{\hat{\ }}{\mathop{i}}\,-12\overset{\hat{\ }}{\mathop{j}}\,\]
Firstly compute the unit vector along the direction of the force.
The unit vector along the direction of the force is the ratio of the direction vector and its magnitude.
The unit vector along the direction of the force is given as follows.
\[\overset{\hat{\ }}{\mathop{n}}\,=\dfrac{5i-12j}{\sqrt{{{(5)}^{2}}+{{(-12)}^{2}}}}\]
Further solve the above equation to find the value of the unit vector.
\[\begin{align}
  & \overset{\hat{\ }}{\mathop{n}}\,=\dfrac{5i-12j}{\sqrt{169}} \\
 & \Rightarrow \overset{\hat{\ }}{\mathop{n}}\,=\dfrac{5i-12j}{13} \\
\end{align}\]
Now multiply the obtained value of the unit vector to the magnitude of the force in order to obtain the value of the vector force.
So, we get the force vector as,
\[\begin{align}
  & \overset{\to }{\mathop{F}}\,=26\times \dfrac{5i-12j}{13} \\
 & \Rightarrow \overset{\to }{\mathop{F}}\,=2\times (5i-12j) \\
 & \Rightarrow \overset{\to }{\mathop{F}}\,=10i-24j \\
\end{align}\]

The force vector obtained is \[\overset{\to }{\mathop{F}}\,=10\overset{\hat{\ }}{\mathop{i}}\,-24\overset{\hat{\ }}{\mathop{j}}\,\]

Additional Information: The force is categorized as a vector quantity, as it has both the quantities, that is, magnitude and the direction. The unit used to represent the force is Newton abbreviated as “N”. Force can both make the object/body move or make the object/body to come to rest.

Note: The things to be on your finger-tips for further information on solving these types of problems are: The units of the given parameters should be taken into consideration while solving the problem.