1. Consider a cube and pick a corner of your preference. From your corner, there are three diagonals along the faces, and one diagonal which travels through the center of the cube. Using vectors, find the angle between any of the face diagonals and the central diagonal. (A useful application of vectors is finding the angles between bonds in molecular geometry.) 2. The angle between two curves at a point of intersection in the xy-plane is defined to be the angle between their tangent lines at that point. Use the dot product to approximate the (acute) angle at which the curves y = cot x and y = tan x first intersect after x = 0. You might find it helpful to graph the functions using e.g. Desmos. 3. A corner reflector is formed by three mutually perpendicular reflecting surfaces. Explain mathematically why a ray of light incident upon the corner reflector (i.e. striking all three surfaces) is reflected back along a line parallel to the line of incidence. Hint: Consider the effect of a reflection on the components of a vector describing the direction of the light ray. 4. If four vectors a, b, c, and d all lie in the same plane, what can you say about (axb)x(cxd)? Be as specific as possible.
1. Consider a cube and pick a corner of your preference. From your corner, there are three diagonals along the faces, and one diagonal which travels through the center of the cube. Using vectors, find the angle between any of the face diagonals and the central diagonal. (A useful application of vectors is finding the angles between bonds in molecular geometry.) 2. The angle between two curves at a point of intersection in the xy-plane is defined to be the angle between their tangent lines at that point. Use the dot product to approximate the (acute) angle at which the curves y = cot x and y = tan x first intersect after x = 0. You might find it helpful to graph the functions using e.g. Desmos. 3. A corner reflector is formed by three mutually perpendicular reflecting surfaces. Explain mathematically why a ray of light incident upon the corner reflector (i.e. striking all three surfaces) is reflected back along a line parallel to the line of incidence. Hint: Consider the effect of a reflection on the components of a vector describing the direction of the light ray. 4. If four vectors a, b, c, and d all lie in the same plane, what can you say about (axb)x(cxd)? Be as specific as possible.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please do problem 3
Transcribed Image Text:1. Consider a cube and pick a corner of your preference. From your corner, there are three
diagonals along the faces, and one diagonal which travels through the center of the cube.
Using vectors, find the angle between any of the face diagonals and the central diagonal. (A
useful application of vectors is finding the angles between bonds in molecular geometry.)
2. The angle between two curves at a point of intersection in the xy-plane is defined to be the
angle between their tangent lines at that point. Use the dot product to approximate the
(acute) angle at which the curves y = cot x and y = tan x first intersect after x = 0. You
might find it helpful to graph the functions using e.g. Desmos.
3. A corner reflector is formed by three mutually perpendicular reflecting surfaces. Explain
mathematically why a ray of light incident upon the corner reflector (i.e. striking all three
surfaces) is reflected back along a line parallel to the line of incidence.
Hint: Consider the effect of a reflection on the components of a vector describing the direction
of the light ray.
4. If four vectors a, b, c, and d all lie in the same plane, what can you say about (axb)x(cxd)?
Be as specific as possible.
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