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Describe the right-hand rule for determining orientation in a 3D coordinate system.
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Fingers point along positive X, curl towards positive Y, thumb points to positive Z.
How does the equation of a circle in 2D extend to describe a sphere in 3D?
By introducing a z-component, forming the equation x^2 + y^2 + z^2 = r^2.
How do you calculate the vector expression 2A - 3B?
Multiply vector A by 2 and vector B by 3, then subtract each component of B from the corresponding component of A.
What is the role of projection in plotting 3D points?
Projection helps in visualizing and understanding how points relate spatially across different planes.
How can you identify if two vectors are parallel in 3D space?
Two vectors are parallel if one is a scalar multiple of the other.
Explain how a single-point plot differs between 2D and 3D.
In 3D, a point is defined by an ordered triple (x, y, z) rather than a pair (x, y) in 2D.
How is the distance formula extended to 3D space?
By incorporating the z-component: sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2).
What transformation do lines in 2D undergo in 3D space?
Lines in 2D become planes in 3D.
What are octants, and how are they formed in 3D space?
Octants are the eight sections of 3D space formed by the three perpendicular planes (XY, YZ, XZ).
How is the XZ plane oriented in a standard 3D coordinate system?
The XZ plane is horizontal, cutting through space like a floor.
Explain the significance of a position vector in 3D.
A position vector, originating from the origin to a point in space, uniquely defines that point's location in 3D.
How can you determine if a triangle in 3D is isosceles using vectors?
By verifying equality among any two of the three side lengths using the distance formula.
What process is used to determine the center and radius of a sphere in 3D?
Completing the square for each variable component in the sphere's equation.
What additional spatial dimension is considered when transitioning from 2D to 3D?
The z-axis, which represents depth.
Describe how to find a unit vector from a given vector in 3D.
Divide the vector by its magnitude.
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