No, the French did not find the Northwest Passage through the establishment of colonies. The search for the Northwest Passage was primarily conducted by European explorers seeking a direct sea route to Asia through the Arctic region. French explorers did establish colonies in North America but did not find the Northwest Passage.
True. All of them were seeking a route to Asia.
Cartier and Hudson were explorers, both seeking what was known as the Northwest Passage. They both went further north in an attempt to find this.
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Frobisher
European explorers first came to British Columbia in the late 18th century in search of the Northwest Passage, a supposed sea route through North America to the Pacific which would facilitate trade with Asia. They were also seeking new lands for settlement, resources like fur and timber, and to establish colonies for their respective countries.
The first European explorers were mainly looking for new trade routes to Asia in order to bypass Islamic-controlled land routes. Additionally, they were seeking to discover new lands, resources, and potential riches to expand their empires and increase their influence in the world.
As late as the 1800's they were still seeking a passage to Asia. Your question needs a little work to be able to answer.
Sebastian Cabot's biggest accomplishment was his exploration of North America and the mapping of its coastlines, particularly during his 1512 expedition. He is credited with being one of the first Europeans to explore the interior of the continent, seeking a northwest passage to Asia. His voyages contributed significantly to the European understanding of the geography of North America and the potential for further exploration and colonization. Additionally, Cabot's work helped lay the groundwork for future explorers and the eventual establishment of trade routes.
Henry Hudson found the passage way to Asia
They were seeking a faster, cheaper, safer, more reliable way to bring goods back from India and the Far East.
Verrazano, Cartier, and Hudson were all explorers who searched for a western sea route to Asia or a passage to the Pacific Ocean. They were seeking a quicker and more direct route for trading goods with Asia.