🟦 13.1 The wonderful world of 3D!
While all Statics problems represent the reality of the 3D world, many can be analyzed in 2D. Others require a 3D approach.
When the geometry requires a 3D approach, we have three analysis approaches to consider.
If the problem is relatively simple, we may decide to conduct three planar two-dimensional analyses (the xy plane, xz plane, and yz plane). For simple problems, this approach is recommended.
For more complex problems, we may decide to apply the mathematical rigor of vector notation. For that reason, this lesson is principally a review of useful 3D vector operations and vector concepts.
When a problem is extremely complex (e.g. the playground in the photograph), you would use software (finite element analysis) for the analysis. It would be too time-consuming to do these types of calculations by hand.
The problems we solve in this class do not require linear algebra or matrix solution methods. Instead, we will use simple algebra (substitution and elimination methods to solve systems of equations). If you'd like to learn more about solving a system of linear equations, check out this link.
❏ A complex playground design
🟦 13.2 Roadmap of the Statics Toolbox for 3D problems
This image provides an overview for our transition from 2D problem-solving to 3D problem-solving.
🟦 13.3 Review: unit vectors
Unit vectors have a length of one (or unity). They are used to communicate the direction of a vector. They do not have units.
When we want to use a unit vector in the x-direction, y-direction, or z-direction, we use i, j, and k. In English, they are pronounced "i-hat," "j-hat," and "k-hat."
Sometimes we want a unit vector that defines some other direction in space. Use a u for this type of unit vector. Also, use two subscripts to indicate the tail-to-head direction of the unit vector.
Note that the order of the subscripts matter.
Example problem:
Points A and B lie 5m apart, as shown.
You need to write the unit vector that defines the inclination of the line of action that is parallel to line AB. You want to go from A to B.
If an angle is given, the components of the unit vector can be expressed in terms of sine and cosine.
If you know the dimensions (as in this problem), it's best to use ratios to define the unit vector.
Since all unit vectors equal unity by definition, you can always check your unit vector by using the 3D version of the Pythagorean Theorem. Here, you'd take sqrt(0.8^2 + 0.6^2) = 1 to verify that you have written a unit vector.
🟦 13.4 Review: magnitude of a vector
Consider the force vector <3,4,5> Newtons.
It's depicted as head-to-tail addition in the interactive model. It shows that a vector in 3D space can always be visualized in terms of two right triangles.
The first triangle lies in the xy plane. You see Fx=3N and Fy=4N, but not Fz (as it's coming directly towards you. You can calculate the hypotenuse of this triangle using the Pythagorean Theorem. We can call the hypotenuse Fxy. It's equal to 5N.
The second right triangle has a base of Fxy=5N and a height of Fz=5N. Its hypotenuse is equal in length to the magnitude of the original vector. That's sqrt(5^2 + 5^2) = 7.071 N. Again, this is a magnitude.
We would need to multiply that magnitude by the proper unit vector to express it as the original <3,4,5> Newton force vector.
🟦 13.5 Review: position vectors
For position vectors, we typically use the symbol r.
In this image, rectangles C, D, and E all measure 3 feet by 5 feet.
The position vector from A to B is <-5,-3,-3> feet. In other words, to get from A to B, you have to move in the negative x-direction, negative y-direction, and negative z-direction.
Try the position vector from G to A on your own. The answer is <5,-2,-3> feet.
Finally, to go from B to G, the position vector is <0,5,6> feet.
🟦 13.6 Review: vector components and resultants
This flipbook provides a review of:
position vectors
vector components and resultants
unit vectors
In 2D, we visualize a bounding box for vector components in a plane.
In 3D, we can do the same kind of thing. We simply need to use a bounding cuboid instead of a bounding box.
We use the bounding cuboid to visualize the three vector components (x-direction, y-direction, and z-direction).
🟦 13.7 Summary of vector symbols and notation
In our 2D problems, we had the luxury of being a little lax in our notation. For instance, we wrote many force vectors, but we didn't go to the trouble of putting a little arrow on top of the signal, because we all knew that force was a vector.
In 3D problems, it's a good idea for us to be a little more formal with our symbols and notation. We will put little arrows on top of our vectors. When we use a magnitude, we can use single or double vertical bars (either is OK):
| F | = the magnitude of the vector F
||F|| = the magnitude of the vector F
Subscripts:
Many vectors have subscripts, such as rAB (a position vector). This can be read as the position "from A to B." Note that rAB is equal to (-1)rBA. You reverse the signs of all three components when you reverse the direction.
Unit vectors:
Use i, j, and k for unit vectors in the x-, y-, and z-directions
Use u for unit vectors in other directions
The magnitude, by definition, is 1.
Other vectors:
use r for position (with 2 subscripts)
use F for force (with 2 subscripts)
use M for moment (with 2 subscripts)
Remember: when we handwrite vectors, we draw arrows on top. When we type vectors, we make them bold instead. And the unit vectors i, j, and k are both bold and italicized.
🟦 13.8 Review: how to compute the dot product of two vectors
Skim this page to review the basics of the dot product.
http://mechanicsmap.psu.edu/websites/A1_vector_math/A1-3_dotproduct/dotproduct.html
Remember: for dot-products, the end result is a scalar.
Remember: for dot-products, the order of the vectors does NOT matter. (Aٜ·B = B·A). That is, the dot product of two vectors is commutative.
Practice until you have the ability to compute dot products quickly, either by hand, or by programming your calculator to do it for you.
🟦 13.9 Review: how to project vectors with the dot product
Peruse this flipbook to see how we can use the dot product to project a vector component onto a line (or an axis).
Vector projections give us a way to determine the vector components that are parallel to the ray defined by the unit vector.
In this example problem, a component of force vector F is projected to the line (axis) OA, through the dot product operation.
🟦 13.10 Review: how to compute the cross-product of two vectors
Read through this overview of the cross-product: http://mechanicsmap.psu.edu/websites/A1_vector_math/A1-4_crossproduct/crossproduct.html.
Most students are taught either Method 1 or Method 2 in the image. Stick with whatever method you were first taught. There's no need to learn the other method.
Remember: for cross-products, the end result is a vector. It's perpendicular to the plane created by the source vectors.
Remember: for cross-products, the order of the vectors is important. AxB ≠ BxA.
If you reverse the order of the operation, then you are reversing the sign of the answer.
Practice until you have the ability to compute cross products quickly. Ideally, use your calculator for this operation. If your calculator doesn't have this functionality pre-built in, program it in yourself.
We will discover applications for the cross-product in Lesson 14.
🟦 13.11 Condensing to two dimensions
A three-dimensional structure that is symmetric (in terms of both geometry and loading) can be condensed to a 2D problem.
For instance, you might try to tip a sibling's chair by exerting force at B and C on the chair with your two hands.
At first glance, a Statics students might try to use vector notation for this type of problem. But more experienced students would quickly notice that System I is equivalent to System II.
As the geometry and loads are symmetric, we can condense to 2D (System III). We project (or elevate) the xy plane of the chair and sum moments about E to determine whether the chair tips.
❏ Some 3D problems can be modeled in 2D
➜ Practice Problems
Problem 0.
Watch at least the first half of this video.
Problem 1.
Problem 2.
Work through Example 2.7.7. Dot Products on https://engineeringstatics.org/dot_products_2D.html.
Problem 3.
Work through Example 2.8.8. 3D Cross Product. Find the cross product of https://engineeringstatics.org/cross-product-math.html.
Problem 4.
Do any additional self-study needed to arrive to class with the ability to work with position vectors, unit vectors, vector components, vector magnitudes, vector projections (dot products), and the cross-product operation.
Problems 5 through 10.
NOT FOR SUMMER 2024. These will be related to 13.11. In Summer 2024, we will do this kind of thing together, in class.