**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.

**Solution :**

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[15] + 6[21] - 2[33]

= -60 + 126 - 66

= -126 + 126

= 0

Hence the given points are coplanar.

After having gone through the stuff given above, we hope that the students would have understood,"How to Prove the Given 4 Vectors are Coplanar"

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