Pre calculus questions. Find magnitude of vector and general question for dot product of two vectors.?

1) To find the magnitude simply use the formula

| a*i + b*j | = SquareRoot( a^2 + b^2)   | This is the square root of the sum of squares of coordinates.
| 3i + 3j | = SquareRoot(3^2 + 3^2) = SquareRoot(9 + 9) = ?18 = 3?2

2)
Let's look at the dot product definition
a . b = |a| * |b| * cos(a^b)
Where |a|, |b| denotes the length of the vectors and cos(a^b) is the cosine of the angle between them.

So, for the dot product to be equals 0, on of the vectors' length must equal zero, or the cosine of the angle between them must me zero.

1) The length of the vector is zero <=> it is a zero vector.
2) The cosine equals zero <=> the angle angle is ?/2  + ?k, k is integer (?/2 is right angle)

So, the dot product of two vectors is zero iff either of the vectors is a zero vector or they are orthogonal (perpendicular).