Question

1) Define conservative forces.
2) value of $\left[\left(\hat{i}\times \hat{j}\right)\times \left(\hat{j}\times \hat{k}\right)\right]=$

Answer 1

Hint
1. We divide forces into two categories first is conservative forces and other is non conservative forces. If the work done by a force during a round trip of a system is always zero, the force is said to be conservative. Otherwise, it is called non conservative.
2. The cross product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ , denoted by $\overrightarrow{a}\times \overrightarrow{b}$ is itself a vector. The magnitude of this vector is $\left|\overrightarrow{a}\times \overrightarrow{b}\right|=ab\sin \theta \hat{n}$ Where, $\theta$ is the angle between vectors. And $\hat{n}$is direction i.e. perpendicular to both the vectors.

Complete Step by step solution
Conservative force: If the work done by a force during a round trip of a system is always zero, the force is conservative. Also, if the work done by a force depends only on the initial and final states and not on the path taken, it is conservative force.
Thus, the force of gravity, coulomb force and all the forces of spring are conservative forces, as the work done by these forces are zero in a round trip. The force of friction is non conservative because the work done by the friction is not zero in a round trip.
The cross product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ , denoted by $\overrightarrow{a}\times \overrightarrow{b}$ is itself a vector. The magnitude of this vector is
$\left|\overrightarrow{a}\times \overrightarrow{b}\right|=ab\sin \theta \hat{n}$
Here, magnitude of vectors is $1$ since all the vectors given are unit vectors and all these vectors are mutually perpendicular hence angle between them is ${90}^o$ .
Therefore, we have
$\left(\hat{i}\times \hat{j}\right)=\hat{k}$ and $\left(\hat{j}\times \hat{k}\right)=\hat{i}$
Hence,
$\left[\left(\hat{i}\times \hat{j}\right)\times \left(\hat{j}\times \hat{k}\right)\right]=\left[\hat{k}\times \hat{i}\right]$
Now $\left[\hat{k}\times \hat{i}\right]=\hat{j}$
Hence, $\left[\left(\hat{i}\times \hat{j}\right)\times \left(\hat{j}\times \hat{k}\right)\right]=\hat{j}$

Note
1. For conservative forces, mechanical energy remains conserved. If the internal forces are conservative then work done by external forces is equal to the change in mechanical energy. If some of the forces are non-conservative then mechanical energy of the system is not conserved.
2. The direction of $\overrightarrow{a}\times \overrightarrow{b}$ is perpendicular to both $\overrightarrow{a}$ and $$\overrightarrow{b}$$ . we use right hand thumb rule to determine the direction of $\overrightarrow{a}\times \overrightarrow{b}$ . in that we have to place our stretched right palm perpendicular to the plane of $\overrightarrow{a}$ and $$\overrightarrow{b}$$ in such a way that the fingers are along the vector $\overrightarrow{a}$ and when the fingers are closed they go towards $$\overrightarrow{b}$$ . the direction of the thumb gives the direction of the arrow to be put on the vector $\overrightarrow{a}\times \overrightarrow{b}$ .
This is known as the right hand thumb rule.