# If u¯=i^-2j^+k^,r¯=3i^+k^ and w¯=j^,k^ are given vectors , then find [u¯+w¯]⋅[(w¯×r¯)×(r¯×w¯)] - Mathematics and Statistics

Sum

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

Concept: Scalar Product of Vectors (Dot)
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