Recent questions tagged vector-identities / GO Mechanical

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GATE2020-ME-2: 26
The directional derivative of $f(x,y,z) = xyz$ at point $(-1,1,3)$ in the direction of vector $\hat{i} – 2 \hat{j} +2 \hat{k}$ is $3\hat{i} – 3 \hat{j} - \hat{k} \\$ $- \dfrac{7}{3} \\$ $\dfrac{7}{3} \\ $ $7$

The directional derivative of $f(x,y,z) = xyz$ at point $(-1,1,3)$ in the direction of vector $\hat{i} – 2 \hat{j} +2 \hat{k}$ is$3\hat{i} – 3 \hat{j} - \hat{k} \\$$-...

go_editor

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go_editor asked Sep 18, 2020

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GATE2020-ME-1: 27
A vector field is defined as ... shell formed by two concentric spheres with origin as the center, and internal and external radii of $1$ and $2$, respectively, is $0$ $2\pi$ $4\pi$ $8\pi$

A vector field is defined as $$\overrightarrow{f}\left ( x,y,z \right )=\dfrac{x}{\left [ x^{2}+y^{2}+z^{2} \right ]^{\frac{3}{2}}}\widehat{i}\:+\:\dfrac{y}{\left [ x^{2}...

go_editor

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go_editor asked Feb 19, 2020

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GATE2019 ME-2: 26
Given a vector $\overrightarrow{u} = \dfrac{1}{3} \big(-y^3 \hat{i} + x^3 \hat{j} + z^3 \hat{k} \big)$ and $\hat{n}$ as the unit normal vector to the surface of the hemipshere $(x^2+y^2+z^2=1; \: z \geq 0)$ ... $S$ is $- \dfrac{\pi}{2} \\$ $\dfrac{\pi}{3} \\$ $\dfrac{\pi}{2} \\$ $\pi$

Given a vector $\overrightarrow{u} = \dfrac{1}{3} \big(-y^3 \hat{i} + x^3 \hat{j} + z^3 \hat{k} \big)$ and $\hat{n}$ as the unit normal vector to the surface of the hemi...

Arjun

28.5k points

Arjun asked Feb 9, 2019

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GATE ME 2012 | Question: 25
For the spherical surface $x^2+y^2+z^2=1$, the unit outward normal vector at th point $\left( \dfrac{1}{\sqrt{2}}, \dfrac{1}{\sqrt{2}}, 0\right)$ is given by $\dfrac{1}{\sqrt{2}} \hat{i} +\dfrac{1}{\sqrt{2}} \hat{j} \\$ ... $\hat{k} \\$ $\dfrac{1}{\sqrt{3}} \hat{i} +\dfrac{1}{\sqrt{3}} \hat{j} +\dfrac{1}{\sqrt{3}} \hat{k}$

For the spherical surface $x^2+y^2+z^2=1$, the unit outward normal vector at th point $\left( \dfrac{1}{\sqrt{2}}, \dfrac{1}{\sqrt{2}}, 0\right)$ is given by$\dfrac{1}{\s...

Andrijana3306

1.5k points

Andrijana3306 asked Mar 19, 2018

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GATE2018-2-28
For a position vector $\overrightarrow{r} = x \hat{i}+y \hat{j} + z\hat{k}$ the norm of the vector can be defined as $\mid \overrightarrow{r} \mid = \sqrt{x^2+y^2+z^2}$. Given a function $\phi =\text{ln} \mid \overrightarrow{r} \mid$, its ... $\dfrac{\overrightarrow{r}}{\overrightarrow{r} \cdot \overrightarrow{r} } \\ $ $\dfrac{\overrightarrow{r}}{\mid \overrightarrow{r} \mid^3} $

For a position vector $\overrightarrow{r} = x \hat{i}+y \hat{j} + z\hat{k}$ the norm of the vector can be defined as $\mid \overrightarrow{r} \mid = \sqrt{x^2+y^2+z^2}$. ...

Arjun

28.5k points

Arjun asked Feb 17, 2018

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GATE2018-1-27
The value of the integral over the closed surface $S$ bounding a volume $V$, where $\overrightarrow{r} = x \hat{i} + y \hat{j}+z \hat{k}$ is the position vector and $\overrightarrow{n}$ is the normal to the surface $S$, is $V$ $2V$ $3V$ $4V$

The value of the integralover the closed surface $S$ bounding a volume $V$, where $\overrightarrow{r} = x \hat{i} + y \hat{j}+z \hat{k}$ is the position vector and $\over...

Arjun

28.5k points

Arjun asked Feb 17, 2018

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GATE2017 ME-2: 2
The divergence of the vector $-yi+xj$ is ________.

The divergence of the vector $-yi+xj$ is ________.

Arjun

28.5k points

Arjun asked Feb 26, 2017

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GATE2017 ME-1: 27
For the vector $\vec{V}=2yz\hat{i}+3xz \hat{j}+4xy \hat{k}$, the value of $\bigtriangledown$. $(\bigtriangledown \times \vec{V})$ is ________.

For the vector $\vec{V}=2yz\hat{i}+3xz \hat{j}+4xy \hat{k}$, the value of $\bigtriangledown$. $(\bigtriangledown \times \vec{V})$ is ________.

Arjun

28.5k points

Arjun asked Feb 26, 2017

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GATE2016-3-53
A point P $(1, 3,−5)$ is translated by $2\hat{i}+3\hat{j}-4\hat{k}$ and then rotated counter clockwise by $90^\circ $ about the $z$-axis. The new position of the point is $(−6, 3,−9)$ $(−6,−3,−9)$ $(6, 3,−9)$ $(6, 3, 9)$

A point P $(1, 3,−5)$ is translated by $2\hat{i}+3\hat{j}-4\hat{k}$ and then rotated counter clockwise by $90^\circ $ about the $z$-axis. The new position of the point...

Arjun

28.5k points

Arjun asked Feb 24, 2017

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GATE2016-3-27
The value of the line integral $\oint_{c}^{ }\overline{F}.{\overline{r}}'ds$ ,where $C$ is a circle of radius $\dfrac{4}{\sqrt{\pi }}$ units is ________ Here, $\overline{F}(x,y)=y\hat{i}+2x\hat{j}$ ... $x-y$ Cartesian reference. In evaluating the line integral, the curve has to be traversed in the counter-clockwise direction.

The value of the line integral $\oint_{c}^{ }\overline{F}.{\overline{r}}'ds$ ,where $C$ is a circle of radius $\dfrac{4}{\sqrt{\pi }}$ units is ________Here, $\overline{F...

Arjun

28.5k points

Arjun asked Feb 24, 2017

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GATE2016-2-26
A scalar potential $\varphi$ has the following gradient: $\bigtriangledown \varphi =yz\hat{i}+xz\hat{j}+xy\hat{k}$ . Consider the integral $\int_{c}^{ }\bigtriangledown \varphi .d\overrightarrow{r}$ on the curve $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$. The curve $C$ ... . The value of the integral is ________

A scalar potential $\varphi$ has the following gradient: $\bigtriangledown \varphi =yz\hat{i}+xz\hat{j}+xy\hat{k}$ . Consider the integral $\int_{c}^{ }\bigtriangledown \...

Arjun

28.5k points

Arjun asked Feb 24, 2017

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GATE2016-1-53
The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X$-$Y$ plane about the vertex $P$ by angle $\theta$ in clockwise direction. If sin$\theta$ = $0.6$ and cos$\theta$ = $0.8$, the new coordinates of the vertex $Q$ are $(4.6, 2.8)$ $(3.2, 4.6)$ $(7.9, 5.5)$ $(5.5, 7.9)$

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X$-$Y$ plane a...

Arjun

28.5k points

Arjun asked Feb 24, 2017

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GATE2015-3-24
Let $\phi$ be an arbitrary smooth real valued scalar function and $\overrightarrow{V}$ be an arbitrary smooth vector valued function in a three-dimensional space. Which one of the following is an identity? Curl$(\phi \overrightarrow{V})$ = $\bigtriangledown$($\phi$ ... $\overrightarrow{V}=0$ Div($(\phi \overrightarrow{V})$ ) = $\phi$ Div$\overrightarrow{V}$

Let $\phi$ be an arbitrary smooth real valued scalar function and $\overrightarrow{V}$ be an arbitrary smooth vector valued function in a three-dimensional space. Which o...

Arjun

28.5k points

Arjun asked Feb 24, 2017

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GATE Mechanical 2014 Set 4 | Question: 44
Consider a velocity field $\overrightarrow{V}=k(y\hat{i}+x\hat{k})$ , where $K$ is a constant. The vorticity, $Ω_Z$ , is $-K$ $K$ $-K/2$ $K/2$

Consider a velocity field $\overrightarrow{V}=k(y\hat{i}+x\hat{k})$ , where $K$ is a constant. The vorticity, $Ω_Z$ , is$-K$$K$$-K/2$$K/2$

Arjun

28.5k points

Arjun asked Feb 19, 2017

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GATE Mechanical 2014 Set 3 | Question: 3
Divergence of the vector field $x_2z\hat{i}+xy\hat{j}-yz\hat{k}$ at $(1,-1,1)$ is $0$ $3$ $5$ $6$

Divergence of the vector field $x_2z\hat{i}+xy\hat{j}-yz\hat{k}$ at $(1,-1,1)$ is$0$$3$$5$$6$

Arjun

28.5k points

Arjun asked Feb 19, 2017

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GATE Mechanical 2014 Set 2 | Question: 3
Curl of vector $\overrightarrow{F}=x^2z^2\hat{i}-2xy^2z\hat{j}+2y^2z^3\hat{k}$ is $(4yz^3+2xy^2)\hat{i}+2x^2z\hat{j}-2y^2z\hat{k}$ $(4yz^3+2xy^2)\hat{i}-2x^2z\hat{j}-2y^2z\hat{k}$ $2xz^2\hat{i}-4xyz\hat{j}+6y^2z^2\hat{k}$ $2xz^2\hat{i}+4xyz\hat{j}+6y^2z^2\hat{k}$

Curl of vector $\overrightarrow{F}=x^2z^2\hat{i}-2xy^2z\hat{j}+2y^2z^3\hat{k}$ is$(4yz^3+2xy^2)\hat{i}+2x^2z\hat{j}-2y^2z\hat{k}$$(4yz^3+2xy^2)\hat{i}-2x^2z\hat{j}-2y^2z\...

Arjun

28.5k points

Arjun asked Feb 19, 2017

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GATE Mechanical 2014 Set 1 | Question: 5
Which one of the following describes the relationship among the three vectors, $\hat{i}+\hat{j}+\hat{k}$ , $2\hat{i}+3\hat{j}+\hat{k}$ and $5\hat{i}+6\hat{j}+4\hat{k}$ ? The vectors are mutually perpendicular The vectors are linearly dependent The vectors are linearly independent The vectors are unit vectors

Which one of the following describes the relationship among the three vectors, $\hat{i}+\hat{j}+\hat{k}$ , $2\hat{i}+3\hat{j}+\hat{k}$ and $5\hat{i}+6\hat{j}+4\hat{k}$ ?T...

Arjun

28.5k points

Arjun asked Feb 19, 2017