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.

Answer 1

 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.