For any two vectors `vec(a) and vec(b)` consider the following statement :
1. `|vec(a)+vec(b)|=|vec(a)-vec(b)|hArr vec(a), vec(b)` are orthogonal.
2. `|vec(a)+vec(b)|=|vec(a)|+|vec(b)|hArrvec(a), vec(b)` are orthogonal.
3. `|vec(a)+vec(b)|^(2)=|vec(a)|^(2)+|vec(b)|^(2)hArr vec(a),vec(b)` are orthogonal.
Which of the above statements is/are correct?

Options

A. `1` and `2` only
B. `1` and `3` only
C. `2` and `3` only
D. `1, 2` and `3`

Answer

B

Explanation

(i) `|vec(a) + vec(b)|=|vec(a)-vec(b)|`
Squaring both the sides
`|vec(a)+vec(b)|^(2)=|vec(a)-vec(b)|^(2)`
or, `|vec(a)|^(2)+ |vec(b)|^(2)+2vec(a).vec(b)=|vec(a)|^(2)+|vec(b)|^(2)-2vec(a) vec(b)`
or, `4vec(a). vec(b)=0`
`rArr vec(a) and vec(b)` are orthogonal.
(ii) `|vec(a)+vec(b)|^(2)=|vec(a)|+|vec(b)|`
Squaring both the sides
`|vec(a)+vec(b)|=(|vec(a)|+|vec(b)|)^(2)`
or, `|vec(a)|^(2)+|vec(b)|^(2)+2vec(a).vec(b)=|vec(a)|^(2)+|vec(b)|^(2)+2|vec(a)||vec(b)|`
or, `vec(a). vec(b)=|vec(a)|. |vec(b)|`
or, `|vec(a)|. |vec(b)|. cos theta = |vec(a)||vec(a)|`
`cos theta =1 rArr theta = 0`
`vec(a) and vec(b)` are parallel, and not orthogonal.
(iii) `|vec(a)+vec(b)|^(2)=|vec(a)|^(2)+|vec(b)|^(2)`
`|vec(a)|^(2)+|vec(b)|^(2)+2vec(a).vec(b)=|vec(a)|^(2)+|vec(b)|^(2)`
`2 vec(a). vec(b)=0`
`rArr vec(a) and vec(b)` are orthogonal.
Statement 1 and 3 are correct.