Limit question involving vectors

[cyt91]

A and B are 2 non-zero constant vectors and |B| =1. If the angle between them is pi/4, find the limit of (|A+xB|-|A|)/x as x approaches 0.

I'm stuck at this question. I've tried using dot product and vector product. But...I don't see the connection ... Any useful hints would be helpful.

Thanks.

 

[dynamicsolo]

It might pay to draw a picture of the vectors A and A + xB ; you can make a triangle, the third leg of which is xB . You can then get a relationship between the length of A and of A + xB from the Law of Cosines. Set up your limit expression and see what happens...

 

[cyt91]

This is my working. I'm still stuck...

https://skydrive.live.com/?sc=photo...751C72E14AD&id=6B041751C72E14AD!159&sc=photos

 

[dynamicsolo]

You're trying to work with vector quantities when the Law of Cosines will let you have scalars. We have

| A + xB |² = A² + x² - 2Ax ( -√2 / 2 ) = A² + x² + (√2) Ax

(we've removed B , since it is a unit vector). The limit we want to evaluate is

$$lim_{x \rightarrow 0} \frac{\sqrt{A^{2} + x^{2} + (√2 \cdot Ax) } - A}{x} .$$

L'Hopital isn't going to do it for this because of the radical, so you want to use "conjugate factors" , which will ultimately let you eliminate the "x" in the denominator and have a ratio the limit of which is not indeterminate.

 

[cyt91]

Got it!

You're right. When in doubt,draw.

Thanks.