Each VLAN is a separate layer-2 network = broadcast domain.
Traffic between layer-2 networks = IP subnets needs to be routed. This is what a router or a layer-3 switch does.
A client (source) sends a routable packet to the (default) gateway, the L3 switch. The packet is encapsulated in an L2 frame addressed to the gateway.
As a router, the switch ...
I guess R3 don't know the "network 1" (subnet between R1 and R2).
When R1 sends the icmp echo request (ping) it will use the IP address of the interface that the packet uses to exit the router as a source, in this case, the IP of the interface connected to "network 1".
from R1, try extended ping using the IP of the interface connected to Network2.
here you ...