Question

Show that if a and b are :
(a). in the same direction then |a + b| = |a| +|b|.
(b). in the opposite direction then |a − b| = |a| + |b|.

Answer 1

Align both vectors along the X-axis of cartesian coordinates. Let the vector $a$ be directed in a positive direction. Then we can write:

$a=(a_x,0,0,\dots,0)\\$

$b=(b_x,0,0,\dots,0)\\$

If vectors are in the opposite direction, then $|a| = a_x>0, |b| = -b_x >0.\\$

$|a-b| = |(a_x-b_x,0,0,\dots,0)|=\sqrt{(a_x-b_x)^2}=\\$

$a_x-b_x=|a|+|b|.\\$

If vectors are in the same direction, then $|a| = a_x>0, |b| = b_x >0.\\$

$|a+b| = |(a_x+b_x,0,0,\dots,0)|=\sqrt{(a_x+b_x)^2}=\\$

$a_x+b_x=|a|+|b|.\\$