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You have the momentum given by the vector: $\vec{p}=m\vec{v}$ $\vec{v}=$ being the velocity vector. Now if you integrate the momentum with respect to velocity then you have the integral: $\int\vec{p} \cdot d\vec{v}=m\int \vec{v} \cdot d\vec{v}$ Where $\vec{v} \cdot d\vec{v}$ is the dot(scalar) product between the two vectors $\vec{v}$ and $d\vec{v}$ ...

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You can get the direction of the field without actually drawing it. The magnetic field of the current through the resistor is not just up or down. The field lines go in a circle around the resistor. You can use the right-hand rule to visualize which the way the lines go around, either clockwise or counterclockwise. If the current flows from $b$ to $a$, ...

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Use the right hand rule. Point your thumb in the direction of the current and your fingers will curl around the wire in the direction of the magnetic field. For current flowing from a to b: Your thumb points down and to the right. Your fingers will be to the left of the loop pointing downward. If you curl them around you will see that the magnetic field ...

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Vectors are perfect for linear motion. But half of mechanics behavior is about angular motion. Also in various cases you can factor a scalar out of a vector, and it tells very convenient things. e.g. amplitude out of the oscillation. Or phase. Energy-wise stuff is then way more easy to right. Also maths objects you get in physics are not only scalar and ...

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I think your confusion is arising from the fact that you are imagining operators as matrices. This is mostly fine, but in this case, the operator itself being a vector is what is causing the confusion - so let me elaborate. ${\bf A}$ is a vector of operators. For example $${\bf A} = \pmatrix{ A_1 \\ A_2 \\ A_3}$$ We can denote this collectively as $A_i$. ...

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on solving the equation the v=0 and u= (-500) according the equation the resolution in not in the quadrant of the resultant so the resolution is negative in this case. i think this is the answer if the second equation is not considered there are infinite possibilities

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The velocity is downward, and the acceleration is downward. Whatever direction you choose, if you start with a velocity of zero the sign of both will be the same (if you throw the feather down, it will decelerate - so the acceleration will the "up". I don't think that is intended here). Whether the floor or the hand is zero in the coordinate system doesn't ...

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Starting from position measurements $x[i]$ taken at times $t[i]$, the total distance travelled is given by $$d[i] = \sum_{j=0}^{i-1}\left|x[j+1] - x[j]\right|$$ or, equivalently $$d[i] = \sum_{j=0}^{i-1}\left|\Delta x[j]\right|$$ where $\Delta x[j] = x[j+1] - x[j]$. Note the absolute values in the sum formulas.

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I am not going to answer your question for you, as that would deprive you from learning the material. I can clear up a few things for you though. The mass is $2kg$ The Force is given as three component vectors in the x, y, z direction ($i,j,k$) It gives you its velocity is three component vectors as well. You are given Force, Velocity($V_i$), Mass, and ...

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