[size=100]i) Every plane vector [math]a^{\rightarrow}=\left(a1,a2\right)[/math] can be wrtten as [math]a^{\rightarrow}[/math] = (a[sub]1, [/sub]a[sub]2[/sub]) = a [sub]1[math]i^{\rightarrow}[/math][/sub] +a[sub]2[math]j^{\rightarrow}[/math] and conversely.[br][/sub]ii) Every space vector [math]a^{\rightarrow}[/math] = (a[sub]1[/sub],a[sub]2[/sub],a[sub]3) can be written as [math]a^{\rightarrow}[/math] = (a[sub]1[/sub],a[sub]2[/sub],a[sub]3) = a[sub]1[math]i^{\rightarrow}[/math]+[/sub]a[sub]2[math]j^{\rightarrow}[/math][/sub]+a[sub]3[math]k^{\rightarrow}[/math][/sub][/sub][/sub][br]and conversely. [br][b]Proof: [/b] i) Here, we have L.H.S. = [img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFcAAAATCAYAAADlAZEWAAADSklEQVRYR+2XWchNURTHf58HJGVImcfyYMw8hEyFFJGpDEVmkTmKMuRFREkyZIiSQsQDL0II5cGQ8iJDGUrxIrzRv9bR/k77fPfuc869deruOnW7Z6211/qvtf5rnTpqp2II1FXMcs0weYA7CnhUBSyvArOqcE/oFbuAPT6lPMCV3X3ASeBjqGdlyt8FLgEnypBvBfQAZgB3gHtl6JQSEYDzgU/A/RiYi4H3vntCwNUFOo2Adp6nMeTSCfFAhwFXgC6lELD38vMZsMVAyApuJ2CpA+gDs7/R8Ud3jIv7FwJuQ7EdtGDKjD9ITJSzDXgYpAWqdrVrVnCXAGuBwXb/GmAH0NHxZzvwArjl+ugDVxmYDbwG3gG3A4PKW/wP0Ab4GTM8GVBQe4GxwO7Y+1BwmwHLgZaWFLX7OaAtsMqp3LnAamC8c988YBqwsCFwo9ZXxntbNjoA3zIgJp5cUUJffL0yQeYl0D/2TpWjxF80br3moaQQcNsDn80H+XLa5odvUCmJkpVcdIZYIvomgasLxFVTgFcWkJyvp5AB5DSqQ22IDXKU5edTa1MlXXyoinErSeIh4Aqw5g61nQfOeChFHTLVaMqNR0NUQ61FErgLbOp3N4ED1oreNSMNUil0Jhi/TXR0t1qAEZjaIt5kpIXnxqsRr//1dIJwET2I/+OnNfAWEMj/j8u5yl5XQASu8xVYZhWSBeAstKBJ/Rjo7PisirwMHLP/vgN9gC+xiEMq1wWzF3ABmASsM64VaJuBnQkFMhw4DgxMAneOZUX8obZYZNm7AUxPUXV5qTwBRjjGNHQ0D/YDp4DRVjFajQ6bXDfgrGdbkMxIQEPJPaJDAfkbGAP0NOpRgWlXjmZRpKOh565i2oGF38wkcPW/jGhbOAL8MN7a5DidF2Ahdj5YR7k6oix12VFbibQLR8Eqhg3GgeJBPdG7+OSPbCpZopt+gIpLVX8IuGmYCDz9/gWoyvW424m2Fn1gqOK9tBAScDVlNcxEDU1yulRDWmDldZQYJVnzod7J6yMiL0eT7KhStf5EbZ/2PtHFAOB6WgMePW0u6wHRVyHBjX+C5ohNJlOiUA0xb9KLUrlCoCmgr7XCnCKBWxhQI0dr4FYwZf8AeAyfFMM6ShUAAAAASUVORK5CYII=[/img] = (a[sub]1[/sub],0) + (0,a[sub]2) = a1(1,0) +a2(0,1) =a [sub]1[math]i^{\rightarrow}[/math][/sub] +a[sub]2[math]j^{\rightarrow}[/math][/sub][/sub][/size][br][sub]= R.H.S. Again, R.H.S. = a [sub]1[math]i^{\rightarrow}[/math][/sub] +a[sub]2[math]j^{\rightarrow}[/math][/sub] = [/sub]a[sub]1[/sub](1,0) +a[sub]2[/sub](0,1) = (a[sub]1[/sub],0) +(0,a[sub]2[/sub]) = (a[sub]1,[/sub]a[sub]2[/sub]) L.H.S. proved. [br][sub]Similarly we can prove the second part of the theorem also.[br][/sub][br][sub][br][/sub]