Vectors parallel to the same plane, or lie on the same plane are called coplanar vectors
The three vectors are coplanar if one of them is expressible as a linear combination of the other two.
We have been given that, \(2\hat i-\hat j+\hat k, \) \(\hat i-3\hat j-5\hat k \), and \(3\hat i-4\hat j-\hat k \).
We can form a relation using these three vectors. Say,
Comparing coefficients of \(\hat i,\hat j\) and \(\vec k\),
We get
2 = x + 3y …(1)
–1 = –3x – 4y …(2)
1 = –5x – 4y …(3)
Solving equations (1) and (2) for x and y.
Equation (1), x + 3y = 2
Equation (2), –3x – 4y = –1
Multiply equation (1) by 3.
x + 3y = 2 [× 3
⇒ 3x + 9y = 6 …(4)
Solving equations (4) and (2),
We get
Put in equation (1), we get
2 = x + 3y
⇒ x + 3(1) = 2
⇒ x = 2 – 3
⇒ x = –1
Substituting x = –1 and y = 1 in equation (3), we get
–5x – 4y = 1
⇒ –5(–1) – 4(1) = 1
⇒ 5 – 4 = 1
⇒ 1 = 1
∵, L.H.S = R.H.S
⇒ The value of x and y satisfy equation (3).
Thus, \(2\hat i-\hat j+\hat k, \) \(\hat i-3\hat j-5\hat k \), and \(3\hat i-4\hat j-\hat k \) are coplanar.
