(a) Use the triangle inequality: ||u + v|| <= ||u|| + ||v|| to prove that
(i) ||u - v|| <= ||u|| + ||v||
(ii) ||u|| - ||v|| <= ||u - v||
(b) If ||u|| = 5 and ||v|| = 3, what are the smallest and largest possible values of ||u - v|| and ||v - u||?
(a)
(i) To prove that ||u - v|| <= ||u|| + ||v||, we can start by using the triangle inequality on the expression ||u - v + v||. Adding and subtracting v to the expression, we get:
||u - v + v|| <= ||u - v|| + ||v||
Since ||u - v + v|| is equal to ||u||, we can rewrite the inequality as:
||u|| <= ||u - v|| + ||v||
Rearranging the inequality, we get:
||u - v|| >= ||u|| - ||v||
This proves that ||u - v|| <= ||u|| + ||v||.
(ii) To prove that ||u|| - ||v|| <= ||u - v||, we can start by using the triangle inequality on the expression ||v - u + u||. Adding and subtracting u to the expression, we get:
||v - u + u|| <= ||v - u|| + ||u||
Since ||v - u + u|| is equal to ||v||, we can rewrite the inequality as:
||v|| <= ||v - u|| + ||u||
Rearranging the inequality, we get:
||u|| - ||v|| <= ||u - v||
This proves that ||u|| - ||v|| <= ||u - v||.
(b)
Since ||u|| = 5 and ||v|| = 3, we can use the results from part (a) to find the smallest and largest possible values of ||u - v|| and ||v - u||.
(i) To find the smallest possible value of ||u - v||, we use the result from part (a)(ii) and substitute the values of ||u|| and ||v||:
||u|| - ||v|| <= ||u - v||
5 - 3 <= ||u - v||
2 <= ||u - v||
Therefore, the smallest possible value of ||u - v|| is 2.
(ii) To find the largest possible value of ||u - v||, we use the result from part (a)(i) and substitute the values of ||u|| and ||v||:
||u - v|| <= ||u|| + ||v||
||u - v|| <= 5 + 3
||u - v|| <= 8
Therefore, the largest possible value of ||u - v|| is 8.
Similarly, we can find the smallest and largest possible values of ||v - u||. Since subtraction is not commutative, we need to consider the order of the vectors u and v.
(iii) To find the smallest possible value of ||v - u||, we use the result from part (a)(ii) and substitute the values of ||u|| and ||v||:
||v|| - ||u|| <= ||v - u||
3 - 5 <= ||v - u||
-2 <= ||v - u||
Therefore, the smallest possible value of ||v - u|| is 2.
(iv) To find the largest possible value of ||v - u||, we use the result from part (a)(i) and substitute the values of ||u|| and ||v||:
||v - u|| <= ||v|| + ||u||
||v - u|| <= 3 + 5
||v - u|| <= 8
Therefore, the largest possible value of ||v - u|| is 8.