Question

For two vectors A and B.A+B =A-B, if and only if??​

Answer 1

`\vec{A} + \vec{B} = \vec{A} - \vec{B}` if and only if
`\vec{B}` is a zero vector.

Step-by-step explanation:

An equation is true if and only if adding the same value to both sides of the equation (the value needs to be compatible) gives an equation that is also true.

Start with the
`\vec{A} + \vec{B} = \vec{A} - \vec{B}`.

This equation is true if and only if
`\left(\vec{A} + \vec{B}\right) + \vec{B} = \left(\vec{A} - \vec{B}\right) + \vec{B}` (
`\vec{B}` is added to both sides of the original equation.)

Vector addition and subtraction are associative. Therefore,
`\left(\vec{A} + \vec{B}\right) + \vec{B} = \left(\vec{A} - \vec{B}\right) + \vec{B}` if and only if
`\vec{A} + \left(\vec{B} + \vec{B}\right) = \vec{A} + \left(- \vec{B} + \vec{B}\right)`, which is equivalent to
`\vec{A} + 2\, \vec{B} = \vec{A}`.

`\vec{A} + 2\, \vec{B} = \vec{A}` if and only if
`\left(-\vec{A}\right) + \vec{A} + 2\, \vec{B} = \left(-\vec{A}\right) + \vec{A}` (
`\left(-\vec{A}\right)`is added to both sides of this equation,) which is equivalent to
`2\, \vec{B} = \vec{0}`.

`2\, \vec{B} = \vec{0}` if and only
`\displaystyle (1)/(2) \cdot \left(2\, \vec{B}\right) = (1)/(2) \cdot \vec{0}`, which is equivalent to
`\vec{B} = \vec{0}`. That is:
`\vec{B}` is the zero vector.

In other words:

`\begin{aligned}& \vec{A} + \vec{B} = \vec{A} - \vec{B}\\ &\iff \left( \vec{A} + \vec{B}\right) + \vec{B} = \left(\vec{A} - \vec{B}\right) + \vec{B} \\ &\iff \vec{A} + \left(\vec{B} + \vec{B}\right) = \vec{A} + \left(- \vec{B} + \vec{B}\right) \\ & \iff \vec{A} + 2\, \vec{B} = \vec{A} \\ & \iff \left(-\vec{A}\right) + \vec{A} + 2\, \vec{B} = \left(-\vec{A}\right) + \vec{A} \\ &\iff 2\, \vec{B} = \vec{0} \\ &\iff (1)/(2) \cdot 2\,\vec{B} = (1)/(2) \cdot \vec{0} \\ &\iff \vec{B} = \vec{0}\end{aligned}`.

Hence,
`\vec{A} + \vec{B} = \vec{A} - \vec{B}` if and only if
`\vec{B}` is the zero vector.