Question

Why vectors cannot be added algebraically?

Answer 1

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`