Question Video: Determining the Plane That Contains a Given Vector Using the Direction Angles of Vectors Mathematics

The direction angles of 𝚨 are 90Β°, 97Β°, and 165Β°. Which of the following planes contains 𝚨? [A] π‘₯𝑦 [B] 𝑦𝑧 [C] π‘₯𝑧

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Video Transcript

The direction angles of the vector 𝚨 are 90 degrees, 97 degrees, and 165 degrees. Which of the following planes contains the vector 𝚨? Is it (A) the π‘₯𝑦-plane, (B) the 𝑦𝑧-plane, or (C) the π‘₯𝑧-plane?

We’re given the direction angles of a vector 𝚨. So, let’s begin by reminding ourselves of what these represent. For a vector 𝚨 with components 𝐴π‘₯, 𝐴𝑦, and 𝐴𝑧 in the π‘₯-, 𝑦-, and 𝑧-directions, respectively, the direction angles of 𝚨 are the angles πœƒπ‘₯, πœƒπ‘¦, and πœƒπ‘§ that 𝚨 makes with the π‘₯-, 𝑦-, and 𝑧-axes. Let’s also remind ourselves of the definition of the direction cosines of the vector 𝚨. These are the cosines of the direction angles πœƒπ‘₯, πœƒπ‘¦, and πœƒπ‘§. We can illustrate a general picture of our individual components diagrammatically as shown.

Now, recalling that in a right angle triangle the cos of an angle πœƒ is the length of the side adjacent to the angle divided by the hypotenuse of the triangle, we see that, in terms of the direction cosines, this translates to the cosine of direction angle πœƒπ‘₯ is equal to the π‘₯-component of vector 𝚨 divided by the norm of vector 𝚨, that is, its magnitude, assuming that this is nonzero, and similarly for the direction cosines of angles πœƒπ‘¦ and πœƒπ‘§.

Now, we’ve been given the direction angles of a vector 𝚨. These are 90 degrees, 97 degrees, and 165 degrees. And we’re asked which of the given planes contains the vector 𝚨. To answer this, let’s consider the direction angle πœƒπ‘₯; that’s 90 degrees. If πœƒπ‘₯ is 90 degrees, then the vector 𝚨 is perpendicular to the π‘₯-axis. And we can illustrate this on our diagram as shown. And now if we consider the direction cosine, the cos of πœƒπ‘₯, the cos of πœƒπ‘₯ is the cos of 90 degrees.

We know that the cos of 90 degrees is equal to zero. And this means that the π‘₯-component of our vector 𝚨 divided by the magnitude of vector 𝚨 is also equal to zero. And so this means that the π‘₯-component of our vector 𝚨 must be equal to zero. And if our π‘₯-component of the vector 𝚨 is equal to zero and our vector 𝚨 is perpendicular to the π‘₯-axis, then our vector 𝚨 must be contained in the 𝑦𝑧-plane.

And so if the direction angles of a vector 𝚨 are 90 degrees, 97 degrees, and 165 degrees, then the vector is contained in the 𝑦𝑧-plane, which is option (B).

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