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Question
Find the value of p, if the vectors `hat"i" - 2hat"j" + hat"k", 2hat"i" -5hat"j"+"p" hat "k" , 5hat"i" -9hat"j" + 4 hat"k"` are coplanar.
Solution
Let the given vectors be
`"a" = hat"i" - 2hat"j" +hat"k" , "b"= 2hat"i" - 5hat"j" +"p"hat"k" , "c" = 5hat"i" - 9hat"j" +4hat"k"`
Given that `bar"a", bar"b" ,bar"c"` are coplanar.
These vectors are coplanar if their scalar triple product is zero.
Therefore, we have
` therefore overline("a") . (bar "b" xx bar"c")` = 0
i.e. ` abs[(1 ,-2 , 1),(2 , -5 , "p"),(5 ,-9 ,4)] =0`
1(-20 +9p) + 2(8-5p) + 1 (-18 + 25) =0
= - 20 + 9p + 16 - 10p - 18 + 25 = 0
-p + 3 = 0
- p = - 3
p = 3
Therefore, the value of p is 3.
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