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(II) (a) Determine the magnitude and direction of the sum of the three vectors $$\vec{\mathbf{v}}_{1}=4.0 \hat{\mathbf{i}}-8.0 \hat{\mathbf{j}} \cdot \vec{\mathbf{v}}_{2}=\hat{\mathbf{i}}+\hat{\mathbf{j}} $$ $$ \vec{\mathbf{v}}_{3}=-2.0 \hat{\mathbf{i}}+4.0 \hat{\mathbf{j}} .(b) \text { Determine } \vec{\mathbf{v}}_{1}-\vec{\mathbf{v}}_{2}+\vec{\mathbf{v}}_{3} $$

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(a) 4.2$315^{\circ}$(b) 5.1$280^{\circ}$

Physics 101 Mechanics

Chapter 3

Kinematics in Two or Three Dimensions; Vectors

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Rutgers, The State University of New Jersey

University of Washington

University of Sheffield

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In physics, rotational dynamics is the study of the kinematics and kinetics of rotational motion, the motion of rigid bodies, and the about axes of the body. It can be divided into the study of torque and the study of angular velocity.

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So for party, we want to find the magnitude and direction of the some of the three vectors. But first, of course, we must take the sum of the three vectors. So the so victor sub one plus vectors up to plus vector said three eyes equal to 4.0 I hat minus eight points there. O j hat plus 1.0. All right hat plus 1.0 I hat and then plus negative 2.0. I had plus 4.0 j hat. Ah weakens. We know that this is going to the unequal 3.0 I hat minus 3.0 j hat. So to find the magnitude of the Somme, we can take the square root of the sum of the squares so that what I mean by that is it's going to be 3.0 squared, plus negative 3.0 squared to the 1/2 power, and this is gonna equal 4.2. That would be the magnitude and in the direction would be equal to arc. 10 of the why component. So negative 3.0 divided by the X component 3.0. Um, this is going to be equal. Thio. Well, this is gonna be equal to 45 degrees. However, here, uh, actually, you always wanted to take the positive positive values and then look at the, um look at the ah vectors themselves. We have a positive x coordinate. However, we have a positive component, but a negative. Why component? So this means that we're going to be in the fourth quadrant. And if we're in the fourth continent, this means that this is going to be 45 degrees below the positive X access. So that would be our direction now for B Ah, we have a vector one minus vector to plus factor three. So here it will be. We can say 4.0 I hat minus 8.0 j hot and then minus 1.0 I hat plus 1.0 J hat. Simon plus negative 2.0. I hacked plus 4.0 j hat and this is equaling 1.0 I hat minus 5.0. I had so again Ah, positive X A negative. Why? Yeah, I had my apologies. Ah, positive X negative. Why? This is gonna be in the fourth quadrant. And at this point, we can say that if we wanted to find the magnitude again Aah! This would be equal to 1.0 squared plus and negative 5.0 squared all to the 1/2 power, and this is equaling 5.1. So this would be the magnitude. And then if you wanted to find the direction this would equal to Arc tanne of again negative of Nick rather 5.0 divided by 1.0 and this is gonna equal 70 degrees. And, uh, this Oh, this would be 70 degrees below positive X access. My apologies, 80 degrees. So, uh, are 10 of 5.0 divided by 1.0 would be 80 degrees below positive x axis. This makes sense because most of our magnitude comes from the UAE component, which means that we're going to have a pre large angle because again ah, sign of angles above 45 degrees are greater than, of course, coastline of angles above 45 degrees. That is the end of the solution. Thank you for watching

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