**How to Show the Given Vectors are Coplanar :**

Here we are going to see how to show the given vectors are coplanar.

If the given vectors are coplanar, then one vector is a linear combination of other two vectors.

Now let us see some examples to understand the method of proving given vector are coplanar with some example problems.

**Question 1 :**

Show that the following vectors are coplanar

(i) i− 2 j + 3k, − 2i + 3j − 4k, − j + 2k

**Solution :**

i − 2 j + 3k = s (− 2i + 3j − 4k) + t (− j + 2k)

Equating the coefficients of i, j and k, we get

1 = -2s , s = -1/2

-2 = 3s - t ----(2)

3 = -4s + 2t ----(3)

By applying the value of s in (2), we get t.

3(-1/2) - t = -2

(-3/2) - t = -2

t = (-3/2) + 2

t = 1/2

Now let us check, if the values of s and t satisfies (3).

If it satisfies the equation (3), we may decide that the given vectors are coplanar otherwise they are not.

-4s + 2t = 3

-4(-1/2) + 2(1/2) = 3

2 + 1 = 3

3 = 3

Since it satisfies the condition, the given vectors are coplanar.

(ii) 5i + 6 j + 7k, 7i −8 j + 9k, 3i + 20j + 5k

**Solution :**

5i + 6 j + 7k = s (7i −8 j + 9k) + t (3i + 20j + 5k)

Equating the coefficients of i, j and k, we get

5 = 7s + 3t -----(1)

6 = -8s + 20t ----(2)

7 = 9s + 5t ----(3)

In order to find the value of "s", we have to multiply (1) by 8 and (2) by 7 and add.

56s + 24t = 40

-56s + 140t = 42

--------------------

164t = 82, t = 1/2

By applying the value of t in (1), we get

7s + 3(1/2) = 5

7s = 5 - (3/2)

7s = 7/2

s = 1/2

Now let us check, if the values of s and t satisfies (3).

9(1/2) + 5(1/2) = 7

(9/2) + (5/2) = 7

(14/2) = 7

7 = 7

Hence the given vectors are coplanar.

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