Vectors - Explain these conditions please?

[K_Swiss]

What conditions must be satisfied by the vectors "u" and "v" for the following to be true?

a) |u + v| = |u - v|
vector "u" is perpendicular to vector "v"

b) |u + v| > |u - v|
0° ? ? ? 90°

c) |u + v| < |u - v|
90° < ? ? 180°

------- Can you please explain to me why these conditions are true? Why is it perpendicular for the first one? Why is less less than 90°, but greater than 0° for question "b"? Why is it less than 180°, but greater than 90° for question "c"?

 

[pka]

What conditions must be satisfied by the vectors "u" and "v" for the following to be true?
a) |u + v| = |u - v| vector "u" is perpendicular to vector "v"
b) |u + v| > |u - v| 0° ? ? ? 90°
c) |u + v| < |u - v| 90° < ? ? 180°

First you must understand that the vectors u & v determine a quadrilateral if they are not collinear. The vectors u + v & u - v are the two diagonals of the quadrilateral.

You must think in terms of diagonals to answer these questions.
||u+v|| is the length of the diagonal interior to \(\displaystyle \theta\) the angle between u & v.
a) What is true of a quadrilateral having diagonals the same length?
b&c) If the diagonals have different lengths.

 

[Couvrey]

I think your teacher wanted you to think of this problem graphically (in other words, draw the pictures). If you think of a vector as an arrow (with a beginning and end) and one end is pointing right (vector u) and the other (vector v) starts where vector u ended pointing straight up. u+v is a new straight vector from the start of vector u to the end of v (sorta like a short cut).

Vector u-v is almost the same as vector u+v but the vector v is flipped and pointing in the opposite direction (the minus v means v is now pointing in the opposite direction). When you draw it out with u and v making a right angle, you should notice that the vector u+v and u-v are the same length that's why |u+v| is equal to |u-v| (in case you didn't know, | | <---- these thingys mean length of a vector).

Now, the less than and greater than stuff.
Greater Than
Let's say you start out with that same picture before where vectors u and v make a 90 degree angle (remember: vector v begins where u ended). And let's also say that we tweek the angle between vectors u and v to make it bigger than 90 degrees (like 110 degrees) so that vector v is no longer pointing straight up but now up and slightly to the right. This made the "short cut" vector u+v longer. And if you pointed vector v in the opposite direction, you would have the vector u-v where v is pointing down and to the left. If you drew it out, you world notice that vector u+v is longer than vector u-v. That's why |u+v|>|u-v| when the angle is bigger than 90 degrees.
Less Than
Draw out that same situation where u and v make a 90 degree angle and then tweek the angle so that its now less than 90 so that v is not pointing straight up but now pointing up and to the left. You'll notice that u+v got smaller and if you pointed v in the other direction to make u-v (where v is pointing down and slightly to the right). u-v will make a longer vector than u+v. That's why |u+v|<|u-v| when the angle is smaller than 90 degrees.

Whew that was a lot of writing this late at night...I hope I was of some help
Remember, drawing stuff out is always a good thing in geometry.
Mr Couvrey LaMirada HS teacher of Algebra, Geometry, and Algebra 2

 

[K_Swiss]

I think your teacher wanted you to think of this problem graphically....

Thanks, I understand it now

 

[pka]

There one problem with the above explanation, the material of the angles is backwards. Given two vectors u & v if \(\displaystyle \theta\) is the angle between them then if \(\displaystyle \left\|{u+v}\right\|>\left\|{u-v}\right\|\) then \(\displaystyle \theta\) is acute and if \(\displaystyle \left\|{u+v}\right\|<\left\|{u-v}\right\|\) then \(\displaystyle \theta\) is obtuse.