# Let Veca = Hati + Hatj + Hatk = Hati and Vecc = C_1veci + C_2hatj + C_3hatk Then Let C_1 = 1 and C_2 = 2, Find C_3 Which Makes Veca, Vecb "And" VeccCoplanar - Mathematics

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" vecccoplanar

2) if c_2 = -1 and c_3 = 1, show that no value of c_1can 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

Concept: Scalar Triple Product of Vectors
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