HOW TO SHOW THE GIVEN VECTORS ARE COPLANAR

About "How to Show the Given Vectors are Coplanar"

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.

How to Show the Given Vectors are Coplanar - Questions

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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