# Why vectors cannot be added algebraically?

May 16, 2014

You actually can add vectors algebraically, but they need to be in unit vector notation first.

If you have two vectors $\vec{{v}_{1}}$ and $\vec{{v}_{2}}$, you can find their sum $\vec{{v}_{3}}$ by adding their components.

vec(v_1) = ahat ı + bhat ȷ
vec(v_2) = chat ı + dhat ȷ
vec(v_3) = vec(v_1) + vec(v_2) = (a + c)hat ı + (b+d)hat ȷ

If you wish to add two vectors, but you only know their magnitudes and directions, first convert them to unit vector notation:

vec(v_1) = m_(1)cos(theta_1)hat ı + m_(1)sin(theta_1)hat ȷ
vec(v_2) = m_(2)cos(theta_2)hat ı + m_(2)sin(theta_2)hat ȷ

Then find their sum normally:

$\vec{{v}_{3}} = \vec{{v}_{1}} + \vec{{v}_{2}}$
vec(v_3) = (m_(1)cos(theta_1) + m_(2)cos(theta_2))hat ı + (m_(1)sin(theta_1) + m_(2)sin(theta_2))hat ȷ