Why vectors cannot be added algebraically?

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Why vectors cannot be added algebraically?

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Answer 1 · 2014-05-16T00:50:01

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

If you have two vectors $\to v_{1}$ $vec(v_1)$ and $\to v_{2}$ $vec(v_2)$ , you can find their sum $\to v_{3}$ $vec(v_3)$ by adding their components.

$\to v_{1} = a ˆ ı + b ˆ ȷ$ $vec(v_1) = ahat ı + bhat ȷ$
$\to v_{2} = c ˆ ı + d ˆ ȷ$ $vec(v_2) = chat ı + dhat ȷ$
$\to v_{3} = \to v_{1} + \to v_{2} = (a + c) ˆ ı + (b + d) ˆ ȷ$ $vec(v_3) = vec(v_1) + vec(v_2) = (a + c)hat ı + (b+d)hat ȷ$

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

$\to v_{1} = m_{1} cos (θ_{1}) ˆ ı + m_{1} sin (θ_{1}) ˆ ȷ$ $vec(v_1) = m_(1)cos(theta_1)hat ı + m_(1)sin(theta_1)hat ȷ$
$\to v_{2} = m_{2} cos (θ_{2}) ˆ ı + m_{2} sin (θ_{2}) ˆ ȷ$ $vec(v_2) = m_(2)cos(theta_2)hat ı + m_(2)sin(theta_2)hat ȷ$

Then find their sum normally:

$\to v_{3} = \to v_{1} + \to v_{2}$ $vec(v_3) = vec(v_1) + vec(v_2)$
$\to v_{3} = (m_{1} cos (θ_{1}) + m_{2} cos (θ_{2})) ˆ ı + (m_{1} sin (θ_{1}) + m_{2} sin (θ_{2})) ˆ ȷ$ $vec(v_3) = (m_(1)cos(theta_1) + m_(2)cos(theta_2))hat ı + (m_(1)sin(theta_1) + m_(2)sin(theta_2))hat ȷ$

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Why vectors cannot be added algebraically?

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