I will provide more details for part a below. If any of the steps are unclear please let me know.

__Detailed Explanation for Exercise #1__

1. The information tells us that is NOT the zero vector (because if was the zero vector, then ).

2. Similarly, the information tells us that is NOT the zero vector (because if was the zero vector, then ).

Now the next part of reasoning may seem a little tricky, but if you think about it enough and ask questions about anything that confuses you, I'm sure you will understand.

3. We now look at the condition . We must ask ourselves, how can the cross product be zero? (At this point you should think about this for yourself before moving on).

There are three possible ways that :
Possibility 1:

is the zero vector. However, we showed in step 2 that

is NOT the zero vector

Possibility 2:

is the zero vector.

Possibility 3:

is parallel to

. But what do we know about the cross product

? We know that it should be PERPENDICULAR to

, so it can't be parallel to

.

So the only possibility that can happen is Possibility 2: .

4. We now know that . Just like we did in step 3, we ask ourselves, how can ? Again we have three possibilities:
Possibility 1:

is the zero vector. However, we showed in step 2 that

is NOT the zero vector.

Possibility 2:

is the zero vector. However, we showed in step 1 that

is NOT the zero vector.

Possibility 3:

and

are parallel vectors.

Since Possibilities 1 & 2 don't happen, Possibility 3 is the truth. Since and are parallel, that tells us that , where is a constant we need to determine.

5. We know that . Our goal in this step is to determine the constant . By taking the dot product on both sides of with we get .

From the given information we know and . Plug these numbers in the above equation to get .

6. We now know , so . Now take the dot product on both sides of with A to get .

From the given information we know . Plugging this in the last equation, we get

It may seem complicated, but if you're patient with yourself and ask questions about the parts that don't make sense you'll get it. Good luck!