# Vector Proof Help

• Nov 17th 2010, 04:43 PM
Lord Voldemort
Vector Proof Help
First off, sorry if this is in the wrong section. I was unsure of where to put it, and if there's a more suitable location please point me to it and I'll ask a mod to move it. Anyways, this appeared on my exam today, and I couldn't figure it out on the exam nor could the people that I spoke to.

a and b are unit vectors.
If a dotted with b = -1, show that a = -b
Also, it did not mention any dimension so the "brute force" method (solving for the general case <x,y,z> and <a,b,c> using the dot product definition) isn't ideal. Also, we were to find a method other than using the = |a||b|cos o formula.

I found a way to prove the opposite, as a vector dotted with itself is simply the length of the vector squared, in this case simply one, so if a = -b then a dotted b = a dotted -a = - (a dotted a) = -1. However, I couldn't find a way to link this logic in the other direction.
• Nov 17th 2010, 06:14 PM
sa-ri-ga-ma
The vector becomes negative when it changes the direction. If a and b are in the same direction, the angle between them is zero. cos(0) = 1. so a.b = ab. when a and b are in the opposirte direction, the angle between them is 180 degrees. And cos(180) = -1. So a.b= abcos(180) = -ab.
• Nov 17th 2010, 07:09 PM
Lord Voldemort
Thanks, but the problem specifically said not to use trig functions.
Any ideas?
• Nov 17th 2010, 09:14 PM
sa-ri-ga-ma
You cannot define the dot or cross product of the vectors without using the trigonometric function.
• Nov 18th 2010, 02:50 AM
HallsofIvy
That's not true. There are several methods of defining the dot product of two vectors. For example, in two dimensions, the dot product of <a, b> and <c, d> is ac+ bd. In higher dimensions it is common to use the dot product to define angles.

Lord Voldemort, look at the dot product of $\vec{a}+ \vec{b}$ with itself. $(\vec{a}+ \vec{b})\cdot(\vec{a}+ \vec{b})= \vec{a}\cdot\vec{a}+ 2\vec{a}\cdot\vec{b}+ \vec{b}\cdot\vec{b}$. Use the fact that $\vec{a}$ and $\vec{b}$ are unit vectors and that $\vec{a}\cdot\vec{b}= -1$. What does that give you and what can you conclude? Remember that this is the dot product of a vector with itself.
• Nov 18th 2010, 05:45 PM
Lord Voldemort
Thanks so much, I see it now.