Engineering Mat...

\(\vec a,\;\vec b,\;\vec c\) are three orthogonal vectors. Given that \(\vec a = \hat i + 2\hat j + 5\hat k\) and \(\vec b = \hat i + 2\hat j - \hat k\), the vector \(\vec c\) is parallel to

If \(\vec F = \left( {{x^2}y + 3z} \right)\hat i + \left( {x{z^3} - 2y} \right)\hat j + {x^2}z\hat k.\) Then the value of grad(div \(\vec F\)) at the point (1, 2, 3) is

The directional derivative of ϕ = 5x²y – 5y²z + 2.5z²x at the point P(1, 1, 1) in the direction of the line \(\frac{{{\rm{x}} - 1}}{2} = \frac{{{\rm{y}} - 3}}{{ - 2}} = {\rm{z}}\)

The vector \(\vec V = \left( {x + y + az} \right)i + \left( {bx + 2y - z} \right)j + + \left( { - x + cy + 2z} \right)k\) is irrotational. Where a, b and c are constants. Find the divergence of the vector \(\vec V\).

The vector \(\vec u\) is defined as \(\vec u = y{\hat e_x} - x{\hat e_y}\), where ê_x and ê_y are the unit vectors along x and y directions, respectively. If the vector \(\vec \omega \) is defined as \(\vec \omega = \vec \nabla \times \vec u\), then \(\left| {\left( {\vec \omega .\vec \nabla } \right)\vec u} \right|\) = ________.

The potential function for the vector field \(\vec F = \left( {4xy + {y^2}z} \right)\hat i + \left( {2{x^2} + 2xyz + {z^2}} \right)\hat j + \left( {x{y^2} + 2yz} \right)\hat k\) is given by

Find f(r) such that \(\nabla f = \frac{{\vec r}}{{{r^5}}}\) and f(1) = 0

Let \(\vec a = \lambda \hat i - 9\hat j - \hat k,\;\vec b = 3\hat i + 3\hat j + \hat k\;and\;\vec c = 4\hat i + 2\hat j + \hat k\). The value of λ for which the vector \(\vec a\) is perpendicular to \(\vec b \times {\rm{\;}}\vec c\) is ________.

A particle moves along the curve x = 3t², y = t³ - 3t² and z = 4t - 6, where t is time. The component of velocity at t = 2, in the direction of (2i + 3j - 5k) is -