Let `veca = hati + hatj + hatk = hati` and `vecc = c_1veci + c_2hatj + c_3hatk` then
1) Let `c_1 = 1` and `c_2 = 2`, find `c_3` which makes `veca, vecb "and" vecc`coplanar
2) if `c_2 = -1` and `c_3 = 1`, show that no value of `c_1`can make `veca, vecb and vecc` coplanar
Solution
`veca = hati + hatj + hatk = hati` and `vecc = c_1veci + c_2hatj + c_3hatk`
Let c1 = 1 and c2 = 2
`vecc = hati + 2hatj + c_3hatk`
For vectors to be coplanar scalar triple product should be equal to zero.
`:. veca*(vecb xx vecc) = 0`
`=> (hati + hatj + hatk)*[hati xx (hati + 2hatj+c_3hatk)] = 0`
`=> (hati + hatj + hatk)*(-c_3hatj + 2hatk) = 0`
⇒ 0 - c3+ 2 =0
⇒ c3 = 2
2) If c2 = –1 and c3 = 1
Let `veca, vecb and vecc` be coplanar
For vectors to be coplanar scalar triple product should be equal to zero.
`:. veca*(vecb xx vecc) = 0`
`(veci + hatj + hatk)*[hati xx c_1hati - hatj + hatk)]`
`[hati xx (c_1hati - hatj + hatk)] = [(hati, hatj, hatk),(1,0,0),(c_1, -1, 1)]`
`= hati(0 - 0) - hatj(1-0) + hatk(-1-0)`
`= -hatj - hatk`
So, here we can see that the value of the vector product of `vecb` and `vec c` does involve `c_1`
Therefore we can say that there is no value of `c_1` can make `veca, vecb " and " vecc` coplanar
