Tässä ovat vihkot, joihin kirjoitan uudet sanat.

Tässä means here, but literally it is the inessive form of tämä (this), so its core meaning is in this. In sentences like this, it is very commonly used to point out something that is right here in front of the speaker:

Tässä ovat vihkot = Here are the notebooks.

So although the literal grammar is connected to this, the natural English translation is usually just here.

Both are possible, but they emphasize different things.

• Tässä ovat vihkot = Here are the notebooks.
  This is a presenting structure: the speaker is showing or introducing the notebooks.

• Vihkot ovat tässä = The notebooks are here.
  This focuses more on the location of the notebooks.

So the given sentence sounds like the speaker is pointing them out.

Because vihkot is plural.

• on = is / singular
• ovat = are / plural

So:

• Tässä on vihko = Here is a notebook
• Tässä ovat vihkot = Here are the notebooks

Vihkot is the nominative plural form, and here it refers to a specific set of notebooks: the notebooks.

• vihkot = the notebooks / these notebooks / the full set in context
• vihkoja = notebooks / some notebooks / an indefinite quantity

Compare:

• Tässä ovat vihkot = Here are the notebooks
• Tässä on vihkoja = There are notebooks here

So vihkot sounds definite and specific, while vihkoja would sound more indefinite.

Finnish does not have articles like the or a/an.

Whether something means a notebook or the notebook, notebooks or the notebooks, is usually understood from:

• context
• word order
• case choice
• whether the thing is already known or being introduced

So vihkot can mean the notebooks in the right context, even though there is no separate word for the.

Joihin means into which or which ... into. In natural English, we usually just say that or which:

vihkot, joihin kirjoitan uudet sanat
= the notebooks that I write the new words into

It is a form of the relative pronoun joka (which / that / who).

Here it refers back to vihkot.

Joihin is the illative plural form of the relative pronoun joka.

The illative case often means into.

That case is used because with kirjoittaa in this kind of expression, Finnish thinks of the words as being written into the notebook:

• kirjoittaa vihkoon = to write in / into a notebook

So:

• vihkoihin = into the notebooks
• joihin = into which

That is why joihin is used instead of some other form.

Because joihin means into which, while joissa means in which.

With notebooks, Finnish normally uses the idea of writing something into them:

• kirjoitan sanat vihkoon = I write the words in/into the notebook

So the sentence uses the same pattern:

• vihkot, joihin kirjoitan uudet sanat

If you used joissa, it would suggest simple location in which, but that is not the normal choice with kirjoittaa in this meaning.

Because Finnish usually does not need subject pronouns when the verb ending already shows the person.

• kirjoitan = I write
• the ending -n already tells you the subject is I

So minä kirjoitan is possible, but it is only used when the speaker wants extra emphasis or contrast.

In neutral Finnish, just kirjoitan is completely normal.

This is about the object case.

• uudet sanat = a total object: the speaker means the new words as a complete, specific set
• uusia sanoja = a partial object: some new words, an indefinite amount, or an ongoing/incomplete action

So:

• kirjoitan uudet sanat = I write down the new words
• kirjoitan uusia sanoja = I write some new words

In the given sentence, the idea is that the speaker writes the new words, not just an unspecified number of words.

Sanat is the plural form of sana (word), and here it is a total object.

In Finnish, a plural total object has the same form as the nominative plural. That is why it looks like a nominative form:

• sana = word
• sanat = words

The adjective agrees with it:

• uusi sana = new word
• uudet sanat = new words

So uudet sanat is a matching adjective + noun pair in the plural.

Then both the noun and the relative pronoun would become singular:

Tässä on vihko, johon kirjoitan uudet sanat.

Changes:

• ovat → on because the subject is singular
• vihkot → vihko
• joihin → johon

So:

• plural: Tässä ovat vihkot, joihin kirjoitan uudet sanat.
• singular: Tässä on vihko, johon kirjoitan uudet sanat.