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Standard XII

Physics

Addition and Subtraction in Unit Vector Notation

If the vector...

Question

If the vector function $\to F =_{x} (3 y - k_{1} z) +_{y} (k_{2} x - 2 z) -_{z} (k_{3} y + z)$ is irrotational, then the values of the constants $k_{1}, k_{2}$ and $k_{3}$ , respectively, are

0.3, - 2.5, 0.5

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0.0, 3.0, 2.1

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0.3, 0.33, 0.5

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4.0, 3.0, 2.0

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Solution

The correct option is B $0.0, 3.0, 2.1$
For irrotational vector:
$\to F, c u r l \times \to F = \to 0$
$∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} (3 y - k_{1} z) & (k_{2} x - 2 z) & - (k_{3} y + z) \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = 0$
$^i [- k_{3} + 2] -^j [0 + k_{1}] +^k [k_{2} - 3] = 0$
So, $k_{3} = 2, k_{1} = 0, k_{2} = 3$
$⎡ ⎢ ⎣ \begin{matrix} k_{1} k_{2} k_{3} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 032 \end{matrix} ⎤ ⎥ ⎦$

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Addition and Subtraction in Unit Vector Notation

Standard XII Physics

If the vector function →F=^ax(3y−k1z)+^ay(k2x−2z)−^az(k3y+z) is irrotational, then the values of the constants k1,k2 and k3, respectively, are

The correct option is B 0.0,3.0,2.1For irrotational vector: →F,curl×→F=→0 ∣∣ ∣ ∣ ∣∣^i^j^k∂∂x∂∂y∂∂z(3y−k1z)(k2x−2z)−(k3y+z)∣∣ ∣ ∣ ∣∣=0 ^i[−k3+2]−^j[0+k1]+^k[k2−3]=0 So, k3=2,k1=0,k2=3 ⎡⎢⎣k1k2k3⎤⎥⎦=⎡⎢⎣032⎤⎥⎦

If the vector function $\to F =_{x} (3 y - k_{1} z) +_{y} (k_{2} x - 2 z) -_{z} (k_{3} y + z)$ is irrotational, then the values of the constants $k_{1}, k_{2}$ and $k_{3}$ , respectively, are