Question

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"u" xx bar"r") xx (bar"r" xx bar"w")]`

Answer 1

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

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

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

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

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

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

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

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

Now, `(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 + 0 = - 12