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The projections of a vector on the three coordinate axis are $6,-3,2$ respectively. The direction cosines of the vector are (A) $6,-3,2$ (B) $\frac{6}{5},-\frac{3}{5}, \frac{2}{5}$ (C) $\frac{6}{7},-\frac{3}{7}, \frac{2}{7}$ (D) $-\frac{6}{7},-\frac{3}{7}, \frac{2}{7}$

In problem 99. The projections of victor on the three coordinate axes are 6 -3 and two. We want to find the direction designs of this, we can express a victory. For example, a that equals The magnitude of eight. Multiplied by the projection of a. And toy six multiplied by I plus the projection of The victor in two G blow by J plus the projection of the victor on that access or key or to blow your bike plus talking. Then we can't find the direction cuisines but a little correction here. This is not a this is a vector in the direction of a because this is not an object. And to get a unit vector in the direction of a. You can say that he had, we just divide by the magnitude of a then just a correction. This is not a You can say that it's w where W is in the direction of can find W head that equals the same direction six I minus three G plus two. K. Want to play it by the magnitude of a divided by the magnitude of W which is square root of six squared plus three squared plus two squared which is seven. To blow it by magnitude away. Now we can cancel magnitude of eight and get the direction designs of the factor W hat, which is the same as a to find because I mean see the eye with the direction of I equals W head dot I Which equals six divided by seven because jaded I equals zero K. Did I equals zero. I did I give you one? Because I in strategy equals W head dot J equals minus three divided by seven. And similarly, we can get design theater Okay, equals W had the key equals two divided by six. We can find the answer as.

Yeah, let's find the scalar in vector productions of beyond A. So we know that the scalar projection is going to be a dot be divided by magnitude today. That's our scalar projection and the vector projection is going to be the same thing, although we're going to square the magnitude and then multiply it times of actor. So with this in mind let's find the projection of beyond A. So first we'll find the the dark power between them. So we're going to use 35 as an example, but we're going to be using this process throughout problems 35 and 40. So we have three times five, which is 15. And then we're gonna be multiplying the next term by zero. So 15 is our product and then we want to know the magnitude of a which is going to be five. So we have 15/5. That gives us three for the scaler And then it's 75 squared is 25. So .6 times are vector, let me 0.6 times three, just 1.8. And then a negative four would give us negative 2.4. So this would be our components for the vector projection

So for us to find these two projections, we just need to plug everything into those formulas they give us in the chapter. So notice how both of them have the dot product of a with B as well as, um, the magnitude of a. So we'll need to go out and find both of those. I'll find the magnitude of a first because that's a little bit easier to do. So we're going to square each of the components Adam together. So negative. Five square plus, um, about 12 12 squared, all square rooted. So that would be 25 plus 1 44 which is 1 69. Just 13. Now to get the dot product So a dotted with B, remember, we're going to multiply each of these components together and then add the results, so we're gonna do negative five times for, and then plus 12 times six. So that would be negative. 20 plus 72 which is 52. Now we just need to plug these in, so over here, it's going to be 50 to over 13, which simplifies down to four. Um, so this is our scalar projection. And then for the factor projection. So again we have 15 or 52 over, uh, will be 13 squares that I just be 13 times 13. And then we have our victor A which is negative 5. 12. So that should simplify down to 4/13, and then we just go ahead and distribute. So that would be negative. 33rd and then 48 13th and then this Here is our sector projection.

In this question, it is told that we have to find the vector projection of. We worked on yogurt, so to find it, we have to use the formula. The formula says that we have to make the dot product of beyond you. And we have to divide by magnitude of use where And then we have to multiply it by You worked also here. Okay. So I'm going to put the value of you and be better. You were 30 years to form a minus one door. This is way better. And you will raise one comma two and delayed by square of magnitude of us. And so you're transformed into this world. One squared plus two. Is where And if that's where we have to do, okay? And we have two multiplied by your weapon. That is one common soldier. Okay? We'll see what happens here further. I can see that if I multiply the dog put up here, then it turns out into going to one minus wanting to do, divide by. we got here five and multiplied by a factor of one or two. So this turns out into zero here and zero multiplied by one comma two. Well, they're here again, zero comma zero. So this is the value or I can say that vector projection or after we on you. Okay? Thank you.