M 273 Montana State University Multivariable Calculus Questions I uploaded for you the assignment and the sections that you have to follow it to answer the

M 273 Montana State University Multivariable Calculus Questions I uploaded for you the assignment and the sections that you have to follow it to answer the assignment. Please make sure to do as same as the steps in sections. Let me know if you’ve any questions. Multivariable Calculus M273
Summer 2020
Discussion 1: 12.1–12.5
Name:
Section:
Please do not hand in. Solutions and further video discussion are posted on D2L.
1. Matter of Shading One way to accurately render three-dimensional objects on a computer screen involves
using the dot and cross products. In order to determine how to shade a piece of a surface we need to determine
the angle at which rays from the light source hit the surface. To determine this angle, we compute the dot
product of the light vector with the vector perpendicular to the surface at the particular point, called the normal
vector.
If the light ray hits the surface straight on, that is, has an angle of 180◦ with the normal, then this piece of the
surface will appear bright. On the other hand, if the light comes in on an angle, this piece of the surface will not
appear as bright.
(a) Suppose the light source is placed directly above the xy-plane, so that the light rays come in parallel to
the vector h0, 0, 1i . At what angle (to the normal) do the light rays hit a triangle bounded by the points
(3, 2, 4), (2, 5, 3), and (1, 2, 6)?
Note that your answer will be an obtuse angle (greater than 90◦ ) because the vertex of the angle is formed
by the tails of the two vectors.
(b) Repeat Problem 1, this time using the region bounded by (3, 5, 2), (3, 3, 1), and (1, 3, 1).
(c) Suppose we are standing above the light source looking down on the xy-plane. Which of these two regions
will appear brighter to us?
Discussion 1: 12.1–12.5
Page 2 of 3
2. Consider the points P = (1, −1, 3), Q = (0, 2, 2), and R = (−3, −2, −1).
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(a) Find the vectors P Q, and P R.
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(b) Find a vector orthogonal to both P Q and P R.
(c) Using the point P and the vector you found in part (b), find an equation of the plane containing the points
P, Q and R.
(d) Using the point R and the vector you found in part (b), find an equation of the plane containing the points
P, Q and R. How does this plane compare to the plane you found in part (c)?
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(e) Find the vectors P Q and QR.
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(f) Find a vector orthogonal to both P Q and QR.
(g) Using the point Q and the vector you found in part (f), find an equation of the plane containing the points
P, Q and R. How does this plane compare to the plane you found in parts (c) and (d) ?
Discussion 1: 12.1–12.5
Page 3 of 3
3. Find the equation for the plane that contains the line L(t) = h2t, 1 − t, 4ti and the point P = (1, −6, 5).

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