SUNY Geneseo Department of Mathematics
Tuesday, February 7
Math 223
Spring 2023
Prof. Doug Baldwin
Problem 2 (intersecting spheres) on problem set 2?
Here’s a sketch of the 2 spheres and their equations, showing how they intersect (Part A of the problem asks you to plot a nicer version of this with Mathematica):
From the picture, you might suspect that the intersection is a circle perpendicular to the x axis. To figure out the x coordinate of that circle, treat the equations of the spheres as a system of simultaneous equations (i.e., a set of equations that must all simultaneously be true), and solve for x. Even though the equations aren’t linear, you can use techniques you learned for solving systems of linear equations, such as subtracting one equation from another:
Solving this system of equations gives you a constant value for x. Finally, you can plug that value back into the original equations in place of x, move it to the right-hand sides (since it’s a constant) to get equations that look like y2 + z2 = k for some constant k. This is indeed the equation for a circle, confirming the guess we made based on the sketch.
Suppose someone launches a hot air balloon. We’ll use a coordinate system whose origin is at the launch site, and whose x, y, and z axes point east, north, and up respectively, to describe the balloon’s movement. In this system, suppose the balloon drifts northeast at a speed of 8 feet per second, and rises at a speed of 2 feet per second.
Throughout this problem, it’s helpful to know that “velocity” is the term for a vector that gives you both something’s speed and the direction it’s moving in. Specifically, speed is the magnitude of the vector, and direction is the vector’s direction.
For the first part of the question, use what you’re given about speeds to find 2 velocity vectors, one horizontal (based on the 8 feet per second northeast) and one vertical (based on 2 feet per second up). For the horizontal velocity, use the fact that “northeast” is 45 degrees from the x and y axes to calculate that the speed in the x direction must be 8 cos(45°) and the speed in the y direction 8 sin(45°). Finally, add the horizontal and vertical velocity vectors to get the total velocity:
For the second part, find the magnitude of the velocity vector (since speed is the magnitude of velocity):
Finally, to see if the balloon clears the hills, use it’s speed in the x direction (which is just the x component of its velocity) to calculate how long it takes to get to the hills, then multiply that time by the vertical speed to see if the balloon rises at least 300 feet in the time it takes to reach the hills:
The balloon rises more than 300 feet, so it does clear the hills.
Based on “The Dot Product and Its Properties” (and more) in section 1.3 of the textbook.
The dot product of two vectors is a scalar.
The formula: ⟨x1,y1,z1⟩• ⟨x2,y2,z2⟩ = x1x2 + y1y2 + z1z2. This formula can also be used with vectors expressed in terms of the standard unit vectors.
Dot products follow algebra laws similar to many for numbers, e.g, they are commutative, scalar multiplication associates with dot product, etc.
Some uses of dot products:
Vectors are “orthogonal” if they are perpendicular. The dot product of 2 orthogonal vectors is always 0.
(These will lead off the schedule for tomorrow’s class.)
How to do projection calculations?
What are direction cosines?
Is there an example of orthogonality?
Dot products and their applications, starting from the questions from today.
Please read (or review) “Using the Dot Product to Find the Angle between Two Vectors,” and “Projections” in section 1.3 of the textbook.