Question

If `|bar"a"| = |bar"b"| = 1,  bar"a".bar"b" = 0, bar"a" + bar"b" + bar"c" = bar"0", "find"  |bar"c"|`.

Answer 1

`bar"a" + bar"b" + bar"c" = bar"0"`

∴ `- bar"c" = bar"a" + bar"b"`

Taking dot product of both sides with itself, we get

`(-bar"c").(-bar"c") = (bar"a" + bar"b").(bar"a" + bar"b")`

∴ `|bar"c"|^2 = bar"a".(bar"a" + bar"b") + bar"b".(bar"a" + bar"b")`

`= bar"a".bar"a" + bar"a".bar"b" + bar"b"bar"a" + bar"b"bar"b"`

`= |bar"a"|^2 + 0 + 0 + |bar"b"|^2    ....[bar"a".bar"b" = bar"b".bar"a" = 0]`

= 1 + 1 = 2     .....`[|bar"a"| = |bar"b"| = 1]`

∴ `|bar"c"| = sqrt2`.