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 𝑥𝑦-plane, (B) the 𝑦𝑧-plane, or (C) the 𝑥𝑧-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 𝐴𝑥, 𝐴𝑦, and 𝐴𝑧 in the 𝑥-, 𝑦-, and
𝑧-directions, respectively, the direction angles of 𝚨 are the angles 𝜃𝑥, 𝜃𝑦,
and 𝜃𝑧 that 𝚨 makes with the 𝑥-, 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 𝜃𝑥, 𝜃𝑦, and 𝜃𝑧. 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 𝜃 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 𝜃𝑥 is equal to the 𝑥-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 𝜃𝑦 and 𝜃𝑧.
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 𝜃𝑥; that’s 90 degrees. If 𝜃𝑥 is 90 degrees, then the vector 𝚨 is perpendicular to the 𝑥-axis. And we can illustrate this on our diagram as shown. And now if we consider the direction cosine, the cos of 𝜃𝑥, the cos of 𝜃𝑥 is the
cos of 90 degrees.
We know that the cos of 90 degrees is equal to zero. And this means that the 𝑥-component of our vector 𝚨 divided by the magnitude of
vector 𝚨 is also equal to zero. And so this means that the 𝑥-component of our vector 𝚨 must be equal to zero. And if our 𝑥-component of the vector 𝚨 is equal to zero and our vector 𝚨 is
perpendicular to the 𝑥-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).
