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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