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Question
If `bar"u" = hat"i" - 2hat"j" + hat"k" , bar"v" = 3hat"i" + hat"k"` and `bar"w" = hat"j" - hat"k"` are given vectors, then find `(bar"u" + bar"w").[(bar"u" xx bar"v") xx (bar"v" xx bar"w")]`
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Solution
`bar"u" + bar"w" = (hat"i" - 2hat"j" + hat"k") + (hat"j" - hat"k")`
= `hat"i" - hat"j"`
`bar"u" xx bar"v" = |(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"v" 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"v") xx (bar"v" xx bar"w")] = |(1, -1, 0),(-2, 2, 6),(-1, 3, 3)|`
= 1(6 - 18) + 1(- 6 + 6) + 0
= - 12 + 0 + 0
= - 12
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