PYQs Based Test - 08 : Vector Identities and Properties

The magnitude of the gradient for the function (𝑥, 𝑦, 𝑧) = 𝑥² + 3𝑦² + 𝑧³ at the point (1, 1, 1) is _________.

If A = 2xi + 3yj + 4zk and u = x² + y² + z², then div(uA) at (1, 1, 1) is _________.

The smaller angle (in degrees) between the planes x + y + z = 1 and 2x - y + 2z = 0 is __________.

The directional derivative of $f\left(x,y\right)=\frac{xy}{\sqrt{2}}\left(x+y\right)$ at (1, 1) in the direction of the unit vector at an angle of $\frac{\pi }{4}$ with y-axis, is given by ______.

For a position vector $\vec{r}=x\hat{i}+y\hat{j}+zk$ the norm of the vector can be defined as $\left|\vec{r}\right|=\sqrt{x^2+y^2+z^2}$. Given a function $\varphi =\ln \left|\vec{r}\right|$, its gradient ∇ϕ is