Я искала книгу не наугад: я помнила, на какой полке видела её вчера.

Искала is imperfective, and that fits because the sentence is talking about the process of searching, not the completed result.

• искала = was looking for / looked for
• нашла = found
• поискала = looked for a while

The point is: I wasn’t searching randomly. The speaker is describing how she searched, not saying she completed the action. So искала is the natural choice.

Because книгу is the accusative singular form, and here книга is the direct object of искала.

For many feminine nouns ending in -а, the accusative singular changes to -у:

• книга → nominative
• книгу → accusative

So:

• Я искала книгу = I was looking for a book/the book

In the Russian past tense, verbs agree in gender and number.

Since the speaker is female, the past-tense forms are feminine:

• я искала
• я помнила
• я видела

If the speaker were male, they would be:

• я искал
• я помнил
• я видел

This is one of the big differences from English: Russian past tense shows gender in the singular.

Наугад is an adverb meaning at random, blindly, or by guesswork.

So:

• искать наугад = to search randomly / to look without knowing
• не наугад = not randomly

In this sentence, it means the speaker was not just guessing where the book might be. She had a specific memory of its location.

Because наугад is a fixed adverb, and it is normally written as one word.

But не with adverbs is often written separately when it simply negates them:

• наугад = randomly
• не наугад = not randomly

So the spelling here is standard.

The colon shows that the second part explains the first part.

So the logic is:

• I wasn’t searching randomly:
• I remembered which shelf I had seen it on yesterday.

In English, you might also express this with because. Russian could also use another structure, but the colon works very naturally when the second clause gives the reason or explanation for the first.

This is a very common question.

• помнить = to remember, in the sense of having something in your memory
• вспомнить = to remember / recall, in the sense of bringing it back to mind at a moment

Here, я помнила means the speaker already had that information in mind while searching.

• Я помнила, на какой полке... = I remembered / I knew from memory which shelf...

If you said я вспомнила, it would suggest a moment of sudden recollection:

• Then I remembered which shelf it was on.

That is a different nuance.

Because this means on which shelf, and with на used for location (where?), Russian uses the prepositional case.

• на полке = on the shelf
• на какой полке = on which shelf

Here:

• полке is the prepositional singular of полка
• какой agrees with полке

Compare:

• на полке = on the shelf
• на какой полке? = on which shelf?
• на какую полку? = onto which shelf? (movement, not location)

So the sentence is about where the book was located, not movement.

Because на какой полке видела её вчера is a subordinate clause, specifically an indirect question.

Russian normally uses a comma before a subordinate clause:

• Я помнила, на какой полке видела её вчера.

It works like English I remembered which shelf I had seen it on yesterday, except Russian punctuation requires the comma very clearly here.

Видела is imperfective, and it is more natural here because the sentence refers to the fact of having seen the book there.

• видеть / видела = to see / saw, in a general or ongoing sense
• увидеть / увидела = to catch sight of, to notice, to see at a particular moment

Here the important idea is not the moment when she first noticed the book, but the remembered fact that it was on that shelf. That is why видела sounds better.

Её refers back to книгу.

Russian often keeps the object pronoun in the subordinate clause even when the noun has already been mentioned:

• Я искала книгу...
• ...на какой полке видела её вчера

That makes the clause complete and clear.

In casual speech, Russian sometimes omits something that is obvious from context, but here её is the normal, standard choice.

Also, you will often see ее instead of её in ordinary writing. Both represent the same word; ё is just more precise.

Russian word order is flexible, and вчера can move around depending on emphasis.

Here, the sentence first gives the important information:

• на какой полке = which shelf

Then it adds the time detail:

• видела её вчера

That makes the shelf the more prominent piece of information.

Other orders are possible, for example:

• я помнила, на какой полке вчера видела её
• я помнила, где вчера видела её

But the version in your sentence is natural and clear.

Yes, but it would be less specific.

• где = where
• на какой полке = on which shelf

So:

• Я помнила, где видела её вчера = I remembered where I had seen it yesterday
• Я помнила, на какой полке видела её вчера = I remembered on which shelf I had seen it yesterday

The version with на какой полке is more precise, because it names the exact kind of place.