If `bar"u" = hat"i" - 2hat"j" + hat"k", bar"r" = 3hat"i" + hat"k"` and `bar"w" = hat"j", hat"k"` are given vectors , then find `[bar"u" + bar"w"]*[(bar"w" xx bar"r") xx (bar"r" xx bar"w")]`

#### Solution

`bar"u" xx bar"r" = |(hat"i", hat"j", hat"k"),(1, -2, 1),(3, 0, 1)|`

= `hat"i"(-2 - 0) - hat"j"(1 - 3) + hat"k"(0 + 6)`

= `-2hat"i" + 2hat"j" + 6hat"k"`

`bar"r" xx bar"w" = |(hat"i", hat"j", hat"k"),(3, 0, 1),(0, 1, -1)|`

= `hat"i"(0 - 1) - hat"j"(-3 - 0) + hat"k"(3 - 0)`

= `-hat"i" + 3hat"j" + 3hat"k"`

`bar"u" + bar"w" = (hat"i" - 2hat"j" + hat"k") + (hat"j" - hat"k")`

= `hat"i" - hat"j"`

`[bar"u" + bar"w"]*[(bar"u" xx bar"r") xx (bar"r" xx bar"w")]`

= `|(1, -1, 0),(-2, 2, 6),(-1, 3, 3)|`

= 1(6 − 18) + 1(−6 + 6) + 0

= −12 + 0

= −12

∴ `[bar"u" + bar"w"]*[(bar"u" xx bar"r") xx (bar"r" xx bar"w")]` = −12