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If vector A = ( ax + 3y + 4 z ) i + ( x - 2y + 3z) j + ( 3x + 2 y - z) k is solenoidal, then a is ______ .
Correct answer is '3'. Can you explain this answer?
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If vector A = ( ax + 3y + 4 z ) i + ( x - 2y + 3z) j + ( 3x + 2 y - z)...
If the vector A is solenoidal, then div A= 0.
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If vector A = ( ax + 3y + 4 z ) i + ( x - 2y + 3z) j + ( 3x + 2 y - z)...
To determine if a vector is solenoidal, we need to check if its divergence is zero. The divergence of a vector A = (Ax i + Ay j + Az k) is given by the expression:
div(A) = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z
In this case, vector A is given as:
A = (ax i + 3y i + 4z i) + (x - 2y j + 3z j) + (3x k + 2y k - z k)
To find the divergence, we need to differentiate each component of A with respect to its corresponding variable:
∂Ax/∂x = a
∂Ay/∂y = 3
∂Az/∂z = -1
Now, let's substitute these values into the expression for divergence:
div(A) = a + 3 - 1
Simplifying, we have:
div(A) = a + 2
Since div(A) = 0 for a vector to be solenoidal, we can set up the equation:
a + 2 = 0
Solving for a, we subtract 2 from both sides:
a = -2
Therefore, the value of a that makes vector A solenoidal is -2.
However, the correct answer given is '3', which contradicts our previous result. It is possible that there is an error in the provided answer or the problem statement. Please double-check the given information to ensure accuracy.
div(A) = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z
In this case, vector A is given as:
A = (ax i + 3y i + 4z i) + (x - 2y j + 3z j) + (3x k + 2y k - z k)
To find the divergence, we need to differentiate each component of A with respect to its corresponding variable:
∂Ax/∂x = a
∂Ay/∂y = 3
∂Az/∂z = -1
Now, let's substitute these values into the expression for divergence:
div(A) = a + 3 - 1
Simplifying, we have:
div(A) = a + 2
Since div(A) = 0 for a vector to be solenoidal, we can set up the equation:
a + 2 = 0
Solving for a, we subtract 2 from both sides:
a = -2
Therefore, the value of a that makes vector A solenoidal is -2.
However, the correct answer given is '3', which contradicts our previous result. It is possible that there is an error in the provided answer or the problem statement. Please double-check the given information to ensure accuracy.
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If vector A = ( ax + 3y + 4 z ) i + ( x - 2y + 3z) j + ( 3x + 2 y - z) k is solenoidal, then a is ______ .Correct answer is '3'. Can you explain this answer?
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