Question

If the vector field F = (x + 2y + az)i + (2x - 3y - Z) j+ (4x - y + 2z) k is irrotational then the value of a is?
(a) -4
(b) 3
(c) -3
(d) 4​

Answer 1

Answer:

D is right options

so 4 is right ans

Answer 2

Concept:

We need to recall the concept of irrotational vector to solve this question.

If curl V=0 , then the vector V is called irrotational vector.

Given:

The vector field F = (x + 2y + az)i + (2x - 3y - Z) j+ (4x - y + 2z) k is irrotational.

To find:

The value of a.

Solution:

Vector field F = (x + 2y + az)i + (2x - 3y - z) j+ (4x - y + 2z) k

where

$v_{1}=x+2y+az\\v_{2}=2x-3y-z\\v_{3}=4x-y+2z$

curl F =      $\begin{vmatrix}i & j & k\\ \frac{d}{dx}&\frac{d}{dy} & \frac{d}{dz}\\ v_{1} & v_{2} & v_{3}\end{vmatrix}$

curl F =      $\begin{vmatrix}i & j & k\\ \frac{d}{dx}&\frac{d}{dy} & \frac{d}{dz}\\ x+2y+az & 2x-3y-z& 4x-y+2z\end{vmatrix}$

curl F =  i(-1+1) -j(4-a)+k(2-2)

curl F =  (4-a)j

for F to be irrotational ,

curl F = 0

(4-a) j = 0

   4-a = 0

      a = 4

Hence, the vector field is irrotational for a =4.

option(d) is correct choice.