Bromination of hexene in presence of UV light or heat
Both products are formed although 3-bromocyclohexene is the major product.

Formation of major product: 3-bromocyclohexene

Under UV light, $\ce{Br2}$ undergoes homolytic splitting to generate $\ce{Br*}$ radicals:

$$\ce{Br2 -&gt;[hv] 2Br*}$$

The formation of 3-bromocyclohexene is an example of substitution of alkanes, which require the free-radical mechanism:



In the first step of the upper mechanism, which is also the rate-determining step, a stable allyl radical is generated, which is stabilized by resonance:



As a result, the activation energy of the first step is significantly lowered.

Formation of minor product: 1,2-dibromocyclohexane

Individual bromine radicals are not electrophilic enough to attack the double bond in the cyclohexene, so the formation of 1,2-dibromocyclohexane requires the ions mechanism, typical for addition reactions (the lower mechanism in the following diagram).



The first step in this mechanism is the rate-determining step. In this step, bromine is ionized, which requires a moderate amount of activation energy, albeit still much higher than the rate-determining step of the upper mechanism.

Conclusion

Therefore, the upper mechanism occurs at a much faster rate than the lower mechanism, which makes the major product 3-bromocyclohexene and the minor product 1,2-dibromocyclohexane.



PS: Many people think that addition reaction is very fast. It is only true in water, where the bromine ion is stabilized by solvation in water.



Disclaimer: The mechanism I used for the addition reaction probably contradicts with your book. However, it does not matter. The point is that an ion is formed which makes the activation energy high.