Three Dimensional Vectors

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In summary, to determine the magnitude of resultant force R, you need to break down each force into its separate components and add them together. The coordinate direction angles α, β, and γ can be determined by finding the magnitude along the x, y, and z axes respectively. In three-dimensional vectors, i, j, and k are used to describe the axes. The magnitude in each direction is found using the Pythagorean theorem. 2-dimensional and 3-dimensional vectors can be visualized in a similar way, with the addition of the z component in 3-dimensions. It may be helpful to think of the axes as a flat piece of paper in 2-dimensions, and adding the z-axis as the aspect
  • #1
I need to determine the magnitude of resultant force R
And, determine the coordinate direction angles α,β, γ of force R

I just can't seem to understand 3-d vectors. I'm fine with 2-d, but this is so different to me. The answers are given in bold I just don't know how to get there. I'd really appreciate any help/explanations. I've asked multiple people for help and they were all confused.
I would ask my professor, but he has a very strong accent and is difficult to understand.

Thanks so much!


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  • #2
In your attachment, the answer is (38.22i + 67.99j + 17.5k). In three dimensional vectors, i, j, and k are simply used to describe the axes. i=x , j=y, k=z . So you have a magnitude of 38.22 along the x-axis, 67.99 along the y-axis, and 17.5 along the z-axis.

It's very similar to 2-d vectors. For example, in 2 dimensional kinematics, a ball shot into the air can easily be broken down into a velocity along the y-axis, and a velocity in the x-axis. Visualizing this, you'll have a ball going up and down, as well as left and right; much like being stuck in a piece of paper, you can go up, down, left, and right.

Adding the z component, you can imagine the ball also being allowed to go in and out of the page. In the diagram, you can see the axes that are labeled x, y, and z. The x and y parts are the same exact axes you would see when you plot something like y=2x+3 . This will essentially be the flat piece of paper that you would work on in two dimensional kinematics with the ball moving around. The z component will be the aspect of coming in and out of that page. (the graph is just conventionally rotated so you can see all three components, x, y, and z all at once. )

There is an imagine to try and help you visually follow. As you can separate the components in the x and y directions in 2-d kinematics, you can do the same here.

the magnitude in the x-direction is how much of the force is along the horizontal (or x-axis)

the magnitude in the y-direction is how much the force is along the vertical (or y-axis)

the magnitude in the z-direction is how much the force is coming into, or out of the page, (or the z-axis)

I apologize if this confuses you further. You seemed to have a good grasp on the 2-d vectors, and wanted to build off of that foundation. If you have any questions or would like more clarification, please feel free to message me. I'll be glad to help.
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  • #3
To find the resulting Force, break down each force into its separate components. the force along x, the force along y, the force along z. now simply add these together.

Fx = Fx,1 + Fx,2

Fy = Fy,1 + Fy,2

Fz = Fz,1 + Fz,2

From these, you should get the answer. (38.22x + 67.99y + 17.5z)

Fx = 38.22
Fy = 67.99
Fz = 17.5

To find the magnitude of the combined Force, in 2-dimensions, you just need to find the hypotenuse.

l F l = √(fx)2 + (fy)2

to find the magnitude in 3-dimensions, it's easily the same concept

l F l = √(fx)2 + (fy)2 + (fz)2

What is a three dimensional vector?

A three dimensional vector is a mathematical representation of a physical quantity that has both magnitude and direction in three-dimensional space. It is typically represented by three values, known as components, which correspond to the coordinates of a point in space.

How are three dimensional vectors represented?

Three dimensional vectors are commonly represented using a column or row matrix, with each component listed in order. They can also be represented graphically as an arrow originating from the origin and pointing towards the coordinates of the vector.

What operations can be performed on three dimensional vectors?

Similar to two-dimensional vectors, three dimensional vectors can be added, subtracted, and multiplied by a scalar. They can also be normalized, which means dividing each component by the vector's magnitude to produce a unit vector. Additionally, the dot product and cross product can be performed on three dimensional vectors to calculate their angle and produce a perpendicular vector, respectively.

What is the difference between a position vector and a displacement vector?

A position vector specifies the coordinates of a point in space relative to an origin, while a displacement vector represents the change in position from one point to another. In other words, a displacement vector has both magnitude and direction, while a position vector only has direction.

How are three dimensional vectors used in real life?

Three dimensional vectors have various applications in science, engineering, and technology. They are used in physics to represent forces and motion, in computer graphics to create 3D models, and in navigation systems to determine location and direction. They are also used in fields such as robotics, aerospace engineering, and video game development.

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