Unit Vector Problem: Find Point of Intersection

[hagobarcos]

Homework Statement

For the equations:

y1 = 1-x^2

y2 = x^2 -1

find the unit tangent vectors to each curve at their point of intersection.

Homework Equations

d/dx (y1) = -2x

d/dx (y2) = 2x

The Attempt at a Solution

After solving for points of intersection between the two equations (-1,0) & (1, 0), I proceeded to ask the derivative for the slope of these points.

The slope at x = 1:
for y1 = -2j

for y2 = 2j


The slope at x = -1:
for y1 = 2j

for y2 = -2j

Next, I divided each resultant vector by the magnitude, (2), to obtain the unit vector.

However, this appears to be incorrect, and I am not sure why.

Attached is a photo:

 

[LCKurtz]

Homework Statement

For the equations:

y1 = 1-x^2

y2 = x^2 -1

find the unit tangent vectors to each curve at their point of intersection.

Homework Equations

d/dx (y1) = -2x

d/dx (y2) = 2x

The Attempt at a Solution

After solving for points of intersection between the two equations (-1,0) & (1, 0), I proceeded to ask the derivative for the slope of these points.

The slope at x = 1:
for y1 = -2j

for y2 = 2j


The slope at x = -1:
for y1 = 2j

for y2 = -2j

Next, I divided each resultant vector by the magnitude, (2), to obtain the unit vector.

However, this appears to be incorrect, and I am not sure why.

Attached is a photo:

Slopes are not vectors. The slopes are 2 and -2 which are scalars. To get a vector along a tangent line of slope 2, figure out a $\Delta y$ and $\Delta x$ such that $\frac{\Delta y}{\Delta x}=2$ and make a unit vector out of $\Delta x i + \Delta y j$.

 

[hagobarcos]

Ahhh. Yes. Of course. Took me a minute to think about it ^.^