# Vectors in the plane Linear Algebra

1. Apr 26, 2010

### Newbatmath

1. The problem statement, all variables and given/known data

Which of the below vectors are A). Orthogonal B.) in the same direction C). in opposite directions

2. Relevant equations

Vectors:

u1 = (1,2) u2 = (0,1) u3 = (-2,-4) u4 = (-2,1) u5 = (2,4) u6 = (-6,3)

3. The attempt at a solution

The text told me how to find if two vectors are orthogonal, (u*v = 0), but I don't know the procedure to use when 6 vectors are involved.

They never mentioned directions at all. :( I'm assuming it just refers to the graph running in either the same or opposite directions?

Newb... (really!)

2. Apr 26, 2010

Since you are asked which of these vectors are orthogonal, it is meant to check this property for every possible pair of vectors in your given set.

Also, "to be in the same direction" means "to be linearly dependent".

3. Apr 26, 2010

### CompuChip

For the first one, that's exactly what they want you to do: find pairs of vectors which are orthogonal. So just take the dot product of every pair you can find and check which ones give 0.

I don't know what you mean by "graph," but you can draw a vector as an arrow. When you draw two vectors like, for example (1, 1) and (2, 2), you will see that the arrows point in the same direction. In other words, the vectors are along a line.
On the other hand, if you draw (1, 2) and (-2, -4) you will see that, although they still lie along the same line, the arrows are pointed exactly opposite to each other. This is what they mean by "same" and "opposite" directions

4. Apr 26, 2010

### Gigasoft

For each vector, determine if it is meets the criteria with respect to either of the other vectors, and note which ones.

The direction of a vector is uniquely determined by the corresponding normalized vector. A normalized vector is an unit vector, and thus has length equal to 1. Two vectors point in the same direction if they, when normalized, are equal. They point in the opposite direction if each component of one of the normalized vectors is equal to the negative of the same component of the other normalized vector. Normalizing a vector means dividing each component by the vector's norm (its length).

(Edit: he beat me to it)