(II) Let

Question

(II) Let

\vec{V} = 20.0 \hat{i} + 26.0 \hat{j} - 14.0 \hat{k}

. What angles does this vector make with the x, y, and z axes?

Accepted Answer

Hey, everyone in this problem, we're told that a car is moving on a three dimensional coordinate system and its velocity is represented by the vector V which is equal to 25 I hat plus 10 J hat minus 15 K hat in meters per second. And we're asked to determine the angles that this velocity vector makes with the XY and Z axis. We're given four answer choices. OK. So for the X, the angle with the X axis, the options are 35.8 degrees or 60 degrees. For the angle with the Y axis 71.1 degrees or 78.5 degrees. And the angle with the Z axis 119 degrees or 60.9 degrees. And then our four answer choices are just different combinations of those options. So let's get started. And we're gonna let theta X theta Y and theta ZB, the angle that the vector makes with CX Y and Z axes respectively. OK. So those are the three values that we're trying to find the X theta Y and theta Z. And recall that we can write the axis themselves, the direction of the axis as I hat J hat and K hat respectively. OK. So I hat points in the direction of the X axis J hat points in the direction of the positive Y axis K hat points in the direction of the positive Z axis. Right now, we also wanna think about what we know about dot products. And so recall that the dot product A dot B is equal to the magnitude of vector, I multiplied by the magnitude of ve vector B multiplied by cosine theta where theta is the angle between those two vectors. So what we're gonna do is we're gonna do this in three parts. OK. So for each of the axis, we're gonna take a dot product of our vector V with the unit vector for that axis. OK. Why don't we do that? We're gonna get the angle between the two. So the angle between the vector and the axis, which is exactly what we want to find. OK. So in order to do this, we're gonna need to know the magnitude of vector V. So let's start there. The magnitude of vector V or call is just gonna be the square root of the sum of squares of the components. So we're gonna have the I A component 25 squared plus the J hat component 10 squared plus the K hat component negative 15 all squared. And we're taking the square root of all of that. And when we do that, we get that, this is the square root of 950. And we're just gonna leave it like that for now. We're not gonna round, we're gonna avoid round off error. Um And there's no need to simplify right now and we're gonna leave it like that and we're gonna use that in our calculations and just round in the final value. So let's start with the X axis. So here we're trying to find the X. So we're gonna take our vector V and we're gonna take the dot product with I hat. OK. So that unit vector pointing in the direction of a positive X A and we know that this is gonna be equal to the magnitude of vector V multiplied by the magnitude of vector IAC multiplied by cosine of theta X. Let's substitute in our values. And we know that the magnitude of vector V is the square root of 950 and lets them, let's leave us more room for the left hand side. OK? So let's start on the left hand side. And what we have on the left hand side is the dot product itself. And we're gonna do that the typical way we do that. OK. So we're gonna multiply the corresponding components and add them together. So for V, OK, the I hot component of V is 25 we're taking the dot product with I hat. So the I hat component of that vector is just one. OK. And then we're gonna add, we're gonna do the same for the other component for vector V. The next component is 10 and we just have I hat for our second vector. So the J hat component is zero. Similarly for vector or for the K hat component vector V has a value of negative 15 A vector I hat has a K hat component of zero. OK. So we have two zeros on the left hand side and on the right hand side, this is where we started. We're getting back to this, we have the magnitude of vector V. So the square root of 950 multiplied by the magnitude of I hat. This is a unit vector. And so it just has a magnitude of one multiplied by cosine of the X and that angle that we're looking for. So on the left hand side, all we have is 25. On the right hand side, we have the square root of 950 multiplied by cosine of theta X. So this tells us that cosine of theta X is equal to 25 divided by the square root of 950. And if we take the inverse cosine, we get that theta X is going to be 35.8 degrees. OK. So we found our first angle and we can go up to our answer choices and compare what we found. So option A and B both, both have theta X being 35.8 degrees. So those are potential answers. Option C and D have 60 degrees, which is not what we found. So we can eliminate those. Let's move to theta Y. Now we're gonna do the exact same thing. But now for the Y axis, so for the Y axis, we're going to take our vector V and now we're gonna take the dot product with J hat because that's the unit vector pointing in the direction of the Y axis. This is gonna be equal to the magnitude of V. The magnitude of J hat multiplied by cosine of theta Y vector V dotted with vector J. We're gonna take the IH components. So 25 multiplied by zero plus the J hat components, 10 multiplied by one plus the K hat components negative 15 multiplied by zero. OK. So if we compare this to the X axis, we can see that the V components are always the same. OK? We have the IH component is 25. The J hat component is 10. The K hat component is negative 15. So those are always the same in this calculation. What changes is the second vector and the components for it. All right, on the right hand side, we still have the magnitude of vector V which is the square root of 950. And just like I hat J hat has a magnitude of one. It is a unit vector. And so we have the square root of 950 multiplied by one multiplied by cosine of the Why? So on the left hand side, this time we have 10, on the right hand side, the square root of 950 multiplied by cosine of theta Y. So cosine of theta Y is going to be equal to 10 divided by the square root of 950. When we take the inverse cosine of this guy, we are gonna get that theta Y is equal to about 71.07 degrees. Let's go back to our answer choices and compare again. Now we have eliminated options, D and D, we have it down to A and B based off of the angle with the X axis in our angle with the Y axis. If we round to three significant digits is 71.1 degrees, which matches with option A but does not match with option B which has 78.5 degrees. OK. So it looks like option A is gonna be the correct answer, but let's move to the Z axis as well. And just double check that we've made no mistakes. So we're looking for theta Z now and again, we're doing the exact same process, but now we are using vector K hat. So for the Z axis, we are looking for V dotted with K hat, this is gonna be the magnitude of V multiplied by the magnitude of khat multiplied by cosine of theta Z. On the left hand side, we have the I hat component of V 25 multiplied by the I hat component of K hat which is zero plus the I hat component of V or sorry, the J hat component of V which is 10 multiplied by the J hat component of K zero plus the K hat component of V negative 15 multiplied by the K hat component of K hat which is one and this is all gonna be equal to the square root of 950 multiplied by one multiplied by cosine of theta Z. And just like it was for all of our other A, the left hand side is gonna simplify the negative 15. And so we end up here with cosine of theta Z is equal to negative 15 divided by the square root of 950. And when we take the inverse cosine this time, we get that theta Z is going to be equal to 119.1 degrees. All right. So we have all three angles that we were interested in finding. And we can see that this corresponds with option A. So the angle with the X axis 35.8 degrees with the Y axis is 71.1 degrees and with the Z axis is 119 degrees. Thanks everyone for watching. I hope this video helps see you in the next one.

(II) Let $\vec{V} = 20.0 \hat{i} + 26.0 \hat{j} - 14.0 \hat{k}$ . What angles does this vector make with the x, y, and z axes?

Key Concepts

Vector Components

Dot Product

Cosine of Angles

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