Answer to Question #100543 in Geometry for Felix

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

If there are two vectors of magnitude a and b then their resultant magnitude is written by

(a² + b² + 2abcos$\theta$)^1/2 where θ is the angle between the two vectors

a) when a and b are in the same direction

θ =0⁰ and as cos0⁰=1

then magnitude of two vectors become $\vert a+b \vert=$ (a²+b²+2ab)^1/2=

=(a+b) = |a| + |b|

b) when a and b are in the opposite direction

θ =180⁰ and as cos180⁰=-1

then magnitude of two vectors become $\vert a-b \vert=$ (a²+b²-2ab)^1/2=

=(a-b) = |a| - |b|