Question

find the value of a, if the vectors i-2j+k and ai-5j+3k and 5i-9j+4k are coplanar.

Answer 1

Answer:

Step-by-step explanation:

answer is 2 obtained by determinant method

Answer 2

We have to find the value of a, If the vectors (i - 2j + k) , (a i - 5j + 3k) and (5i - 9j + 4k) are coplanar.

What is coplanar ?

Coplanar vectors are the vectors which lie one the same plane.

Condition of coplanar :

• If three vectors A , B and C are coplanar vectors , then the their scalar triple product must be zero. i.e., A.(B × C) = 0.
• If there are many vectors are in 3d space linearly independent then these are coplanar vectors.

here here, A = (i - 2j + k)

B = (a i - 5j + 3k)

and C = (5i - 9j + 4k)

according to question,

A , B and C are coplanar so, A.(B × C) = 0

$\left|\begin{array}{ccc}1&-2&1\\a&-5&3\\5&-9&4\end{array}\right|=0$

⇒1(-5 × 4 - 3 × -9) - (-2)(a × 4 - 5 × 3) + 1(a × -9 - 5 × -5) = 0

⇒7 + 2(4a - 15) + (-9a + 25) = 0

⇒7 + 8a - 30 - 9a + 25 = 0

⇒-a + 2 = 0

⇒a = 2

Therefore the value of a is 2.