Statics

Lesson 14

3D Statics, Part II

🟦 14.1 Vector symbols and notation

🟦 14.2 Unit vectors

🟦 14.3 Position vectors

🟦 14.4 Position vectors and unit vectors

🟦 14.5 How to compute the dot product of two vectors

🟦 14.6 How to project vectors with the dot product

🟦 14.7 How to illustrate moment as a double-headed arrow

🟦 14.8 Sign conventions for double-arrow moment vectors

🟦 14.9 Uses of the word "moment" in 3D problems

🟦 14.10 Example problem

➜ Practice Problems

🟦 14.1 Vector symbols and notation

In 2D problems, we had the luxury of being a little lax in our vector notation. For instance, we generally didn't go to the trouble of putting a little arrow on top of our force vectors. This is OK, because everyone who studies Statics knows that force is a vector.

In 3D problems, we don't want to mix up vectors and magnitudes, so it's a good idea to be a little more formal with our vector symbols and notation. For example, it's helpful to put little arrows on top of symbols for vectors when we are writing by hand. When typing (e.g. this website), we make the vector boldface instead of drawing the arrow on top.

Types of vectors in 3D Statics:

i, j, and k are unit vectors in the x-, y-, and z-directions
u is used for all other unit vectors (requires 2 subscripts)
r is used for position vectors (requires 2 subscripts)
F is used for force vectors (subscript designates location)
M is used for moment vectors (subscript designates location or axis)

Magnitude of a vector:

To express the magnitude of a vector, use single or double vertical bars:

| F | = ||F|| = the magnitude of the vector F

Subscripts for F:

Use a single subscript to specify the location of the force:

F_B = the force at B

Subscripts for M:

Use a single subscript for moment at a point, and two subscripts for moment about an axis between two points:

M_A = the moment at A

M_AG = the moment about line AG

Subscripts for u and r:

Two types of vectors require 2 subscripts: r and u. The position vector rAB tells us the change in position from A to B. Note that rAB is simply equal to (-1)rBA. This means that you could write rBA by reversng the signs of all three components of rAB.

Should I show units with my 3D Statics vectors?

What a silly question. Of course you should show units!

You can incorporate units in two different ways:

rAB = <5, 3, 1> m

rAB = <5m, 3m, 1m>

All vectors require units, with one exception: unit vectors. Unit vectors are unitless by definition.

🟦 14.2 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.

These are pronounced as "i-hat," "j-hat," and "k-hat."

When typed, they are bold and italicized.

When handwritten, they wear pointed hats.

Sometimes we want a unit vector that defines some other direction in space. Use u for this type of unit vector. Also, use two subscripts to indicate the tail-to-head direction of the unit vector.

Important: note that the order of the subscripts matter.

Example problem 1:

Points A and B lie at an inclination of 28 degrees, 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.

Remember that you can use the unit circle and Pythagorean Theorem to check your work. (cos^2 + sin^2 = 1^2).

Example problem 2:

Points A and B lie 5m apart, as shown.

As before, you want to the unit vector that defines the line of action from A to B.

Since we know the dimensions, use ratios. You shouldn't calculate the angle when ratios are provided.

In this example, you'd take sqrt(0.8^2 + 0.6^2) = 1 to verify that you have written a unit vector.

🟦 14.3 Position vectors

Position vectors measure the distance between two points.

We typically use the symbol r for position vectors. The r stands for radius (you can create a sphere using the coordinates of one point and a position vector to a second point).

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.

Practice the position vector from G to A on your own. The answer is <5,-2,-3> feet.

Also practice the position vector from B to G. This one is <0,5,6> feet.

🟦 14.4 Position vectors and unit vectors

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).

Example problem 3:

In 3D space, point A lies at <2, 0, 3> inches and point B lies at <0, 6, 0> inches.

We compute the length AB using the 3D Pythagorean Theorem.

Then, the unit vector is written by dividing the position vector by the magnitude of the length.

Remember: since all unit vectors equal unity by definition, you can always check your unit vector by using the 3D version of the Pythagorean Theorem. Make sure it has a length of 1.

🟦 14.5 How to compute the dot product of two vectors

Skim this page to review the basics of the dot product. We use the dot product to determine how much of a given vector points in the direction of another vector. This is why when you dot two perpendicular vectors, you receive an answer of zero.

Key take-aways:

For dot-products, the end result is a scalar.
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 a few problems until you have the ability to compute dot products quickly, either by hand, or by programming your calculator to do it for you.

🟦 14.6 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.

Here is a model for how to write out these calculations. (It's the same problem.)

🟦 14.7 How to illustrate moment as a double-headed arrow

The curly arrow symbol for moment is useful in 2D projections, when the moment is about the axis that is perpendicular to the screen.

For other views -- and in 3D illustrations, it's best to use a double-arrow vector instead.

Moment has always been a vector, because it has both a magnitude and direction.

The "direction" of a moment vector is just a bit more abstract than the direction of a force vector. It's a reference to the axis about which there is a tendency to rotate.

We use the right-hand rule to help us convert double-arrow notation to curly-arrow notation and vice versa.

❏ Double-arrow notation

To convert a double-arrow to a curly arrow:

Point your right thumb in the direction of the double-arrow.
Draw a curly-arrow per the curl of your (right hand) fingers.

To convert a curly-arrow to a double-arrow:

Curl your (right hand) fingers to match the curly arrow.
Draw a double-arrow in the direction your right thumb is pointing.

Moments have x, y, and z components in the same way that forces have components.

For the moment components, please use the double-headed vector.

We can illustrate the x, y, and z components of a moment using the idea of the bounding cuboid.

❏ Components of moment vectors

🟦 14.8 Sign conventions for double-arrow moment vectors

This image depicts forces (single vectors) and moments in 3D space.

When the double-arrow points in the +x direction, we say it's a positive moment about x.

When the double-arrow points in the -y direction, we say it's a negative moment about y.

❏ Positive and negative moments

For positive, you can think either 'thumb in the + direction' or counterclockwise (if the reference axis is coming out of the screen towards you).For negative, you can think either 'thumb in the - direction' or clockwise (if the reference axis is coming out of the screen towards you).

🟦 14.9 Uses of the word "moment" in 3D problems

In a 3D problem, we say moment at a point when we want to calculate the three moment components (Mx, My, and Mz).

Here, we see the moment about node A.

❏ Moment at a point

When we say moment about an axis, we only want a moment component (or projection) on a particular axis.

You can dot a moment vector with a unit vector to project it to any axis.

That is:

❏ Moment about an axis

Remember, in both 2D and 3D problems, we use the word "moment" for three different concepts:

(1) An applied moment is a load or input (information that is given to you in the problem statement).

(2) A reacting moment is the reaction that is needed for static equilibrium (what you want to solve).

(3) A moment summation is an operation (i.e. "the sum of the moments about an axis equals zero"). It's the equation of equilibrium.

In Lesson 16, we'll actually learn a fourth way to use the word moment (internal moments). Stay tuned!

❏ Three related concepts

🟦 14.10 Example problem

In this example problem, we have a heavy plate (800 pounds) that supports two loads (PD and MC). It's supported by cables at E and H, and a 3D pin connection at G.

First, we tally our unknowns. The reaction at E and H are vertical only, so the first two unknowns are Ez and Hz.

The pin support has the ability to generate three reactions (x-, y-, and -z). Since there are no x- or y- loads in the system, those reactions are zero. The vertical reaction at G is our 3rd unknown, Gz.

In sum, there are 3 reactions we need to solve.

❏ Example problem

Since all three unknown forces point in the z-direction, summing forces with respect to z won't help us solve anything.

We strategically decide to sum moments about the axis defined by points G and E. We pick that axis because two of the three unknowns are coincident with that axis.

Solving the moment equation, we determine that the cable at H exerts an upward reaction force of 650#.

❏ Partial solution (xz planar projection)

➜ Practice Problems

Problem 1.

Write the unit vector that describes the line of action from A to B.

Rectangles C, D, and E all measure 2m by 4m.

C and D lie in the xy plane.

E is parallel to the xz plane.

Problem 2.

Vector B has been created by rotating vector A 30 degrees about z.

Write out the components of Vector B. Express your answers as fractions and radicals.

Bx = ?

By = ?

Bz = ?

Problem 3.

This is a continuation of the previous problem.

Vector C has been created by rotating Vector B (which lies on b-b) 45 degrees about line d-d (which is perpendicular to line b-b).

What are the components of Vector C? Express your answer in fractions and radicals.

Cx = ?

Cy = ?

Cz = ?

Problem 4.

This Vector C is the same one from the previous problem. What are the vector components in the x'y'z coordinate system? (The x'y' axes are rotated 30 degrees about z compared to the xy axes.)

Cx' =

Cy' =

Cz =