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Let $\vec{E}=2 \hat{\imath}+3 \hat{\jmath}$ and $\vec{F}=2 \hat{\imath}-2 \hat{\jmath} .$ Find the magnitude of a. $\vec{E}$ and $\vec{F}$ b. $\vec{E}+\vec{F}$ c. $-\vec{E}-2 \vec{F}$

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In the given problem. There are two factors. Yeah, and be. And both of these vectors are making an angle. L far with a horizontal. And here. Well, if I was given as attentively so we can take components off these directors, the components will be given ass like this. Yeah, this will be Yeah. Course l for this will be a sorry business. Be course Emperor and the component. No work, Sam. Both of the components in the same direction. And these with me? Yeah. Sign Alpha and be sign up. For now, I have to find the resultant vector Lee, which is given us you minus B. So if I have to find the magnitude off resultant using component matter, I will have to subject that companies it will become minus equal. Sell for minus being core. Sample the horizontal component DX and divide the vertical component will be. Yeah, Sinaloa from minus b. Sign alpa. So the vertical component may become zero, but the horizontal component will become minus two. A cause l for because as far as magnitude is concern, both are having both energy are having same magnet. You would ask four centimeter. So finally the resultant off this director, the the magnitude of the resultant factor is minus two a course l for means minus pool into 410 centimeter. I mean, you co sign off anger. 10 degree means no se minus scared. Centimeter in tow. Call sign. Oh, anger 10 degree. Which comes out to be 0.9. Yeah, for each. So their product will be approximately seven point gate 78 centimeter. They're negative. Sign Li presents that the resultant is left war. This will be the result in and this is leftward means along negative x axis. So if we look for the pure magnitude only Mrs seven wind it it centimeter.

Okay, so first couple here ask us about a vector B divided by some magnitude. D So what happens in D is positive? Well, the direction of B doesn't change. It's still in the positive. Why direction? The magnitude d there were dividing by doesn't change the direction. If it's positive, it's negative. It flips the direction So d is negative. We know what the vector pointing in the negative y direction. Um, may not be. I'm sorry. A and B are perpendicular, so a dot B equal zero again be over. D is in the wind direction, so a dot be over D. The d doesn't change anything here because it's not a vector. This product is still zero. So a cross be we use the right hand rule. We see this. This goes in the positive direction by the same rule. If we flip it and we do be crossed a, we always get the opposite. So now we're going negative Z direction. So since there's a 90 degree angle between A and B ah, the cross product, we end up with just a magnitude that is the product of the magnitudes of the individual vectors. Because it's a would be Abie signed 90 and signed 90 is one. So our magnitude is just a B same thing for H It doesn't matter if you flip the cross product or magnitude will still be a B I, um Again, if d is positive, then we have dinner Man two turns into a B over D and that vector points in the positive Z direction because he does not change the direction which the vector points its case.

All right, So this question mixes. Look back, probably and give you some numbers. And then we have these components. All right, so we know that these there are two vectors Vector A and Vector B, and you can see there pointing these directions. And what we have to do is Advil together. And so, in order to do this first we have to split A and B into their X and Y components. So let's do that. So in green, here are the X component for a look. A little something like that. And then here with orange on the y component. And so, as you can see, it's pretty simple, just like that. All right, so that's a and similar thing can be done with B. This time I'll use, let's say, red for the X component. Look something like that. And then let's use, uh, blue for the White Book so it will look something like that. So now we have split the's A and B vectors into X and Y components. We can have an easier time adding them because we just add the individual components. So one thing to know is that since a and B are both of length 4.0 centimeters and we know that the angle they make with E X axis is the same, like their direction is the same. Well, then, that means that the magnitude of the green a X is going to be the same magnitude of the vector BX right, however there in opposite directions. So the fact that matter is, they'll actually, if you added together, don't cancel each other out. So now all we have to worry about is adding together a why and b y. And so, in order to do that, well, we just do the simple tip to tail method use ad D B Y vector here. As you can see, all we have to do now is find the magnitude of a why and b y and Adam together. So how do we do that? Well, we know that the high pot news of this sort of right angled triangle here is 4.0 centimeters and one other. The angle here's 10 degrees. And so since we know that science data equals two opposite overhype pot news, well, then we know that the opposite side, which in this case will equal to eight. Why, right, since this is the opposite side Anyways, we know the opposite eagle stay Why? And it'll equal to the high pot News time science data. Right, cause already when you just multiply both sides by had they have partners. So we have the eight y equals to 4.0 centimeters time sign 10 degrees and we can figure out that that equals two 0.69 approximately 0.69 centimeters. And so since we also know that the magnitude of a Y equals to the magnitude of B Y, figure out to the magnitude of B Y is also going to be equal to 0.69 centimeters pointing downwards. And so we know that the final vectors see this is C, by the way, is going to be have a magnitude off balm 0.39 approximately a magnitude off 1.39 centimeters and in the direction is down directly downwards. So there's your vector C