Couple of Calc III questions - Vectors, Continuity

[1MileCrash]

Homework Statement

Hey guys, I have two separate questions.

1.)

I am asked for a unit vector pointing from P = (1,2) to Q = (4,6)

In physics, every vector I've ever worked with started at the origin, so these feel weird.

I initially thought that it would simply be 3i + 4j, the differences of each coordinate. However, my book gives the answer as (3/5)i + (4/5)j, they divided each number by sqrt(3^2 + 4^2). But why? Why specifically divide by that? If I understand correctly, if you divide both numbers by anyone number, the vector will still point the same way because you just make a similar triangle. Why divide by vector length?

2.)

Limits and Continuity has been very confusing for me.

I am asked to show that f(x,y) does not have a limit as (x,y) -> (0,0), and it says (Hint: use the line y = mx)

The function is (x^2 - y^2)/(x^2 + y^2). How do I use y = mx?

Homework Equations

The Attempt at a Solution

Looking for more of a explanation to help me better understand, not a solution.

 

[Dick]

1) you divide by the length because they asked for a 'unit vector'. 3i+4j points in the right direction but it doesn't have length 1. 3i/5+4j/5 does. 2) Just put y=mx into the function you are trying to find the limit of and simplify.

 

[1MileCrash]

1) you divide by the length because they asked for a 'unit vector'. 3i+4j points in the right direction but it doesn't have length 1. 3i/5+4j/5 does. 2) Just put y=mx into the function you are trying to find the limit of and simplify.

Thank you!