How to Prove the Given 4 Vectors are Coplanar ?
Here we are going to see how to prove the given 4 vectors are coplanar.
To prove the given 4 vectors are coplanar, we have to form three vectors using those vectors. Then we have to check whether there is any linear relationship.
Let us look into a example problem to understand the concept much better.
Question 1 :
Show that the points whose position vectors 4i + 5j + k, − j − k, 3i + 9j + 4k and −4i + 4j + 4k are coplanar.
Let OA vector = 4i + 5j + k
OB vector = − j − k
OC vector = 3i + 9j + 4k
OD vector = −4i + 4j + 4k
AB vector = OB vector - OA vector
= (-j-k) - (4i + 5j + k)
= -4i -j - 5j - k - k
= -4i -6j - 2k
AC vector = OC vector - OA vector
= (3i + 9j + 4k) - (4i + 5j + k)
= 3i - 4i + 9j - 5j + 4k - k
= -i + 4j + 3k
AD vector = OD vector - OA vector
= (−4i + 4j + 4k) - (4i + 5j + k)
= -4i - 4i + 4j - 5j + 4k - k
= -8i - j + 3k
-4i -6j - 2k = s(-i + 4j + 3k) + t(-8i - j + 3k)
-4 = -s - 8t -------(1)
-6 = 4s - t -------(2)
-2 = 3s + 3t -------(3)
Multiply the (1) by 4 and add (1) + (2)
-4s - 32t + 4s - t = -16 - 6
-33t = -22
t = 2/3
Applying the value of t in (1)
-s - 8(2/3) = -4
-s - (16/3) = -4
-s = -4 + (16/3)
-s = (-12 + 16)/3
-s = 4/3
s = -4/3
By applying the value of s and t, we get
-2 = 3(-4/3) + 3(2/3)
-2 = -4 + 2
-2 = -2
Hence given vectors are coplanar.
By taking determinants, easily we may check whether they are coplanar or not.
If |AB AC AD| = 0, then A, B, C and D are coplanar.
= -4[12+3] + 6[-3+24] - 2[1+32]
= -4 + 6 - 2
= -60 + 126 - 66
= -126 + 126
Hence the given points are coplanar.
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