Vectors in Space_Chap. 2 Flashcards
What’s the result of a cross-product and what calculations is it useful for?
It is a vector that is orthogonal to each vector involved in the cross-product.
It’s useful for:
*distance calculations since the shortest distance is the line orthogonal to both objects.
*calculating the area of a parallelogram, which can be divided by the magnitude of a side to get the distance between 2 lines.
*volume as part of the triple scalar product
*Torque = Fxr
What’s a cylinder?
It is a set of lines parallel to a given line passing through a given curve
What’s the cylindrical coordinate system?
It is a way to describe a location in space with an ordered triple (r, θ, z), where (r, θ) represents the polar coordinates of the point’s projection in the xy-plane, and z represents the point’s projection onto
the z-axis
What are direction angles?
They are the angles formed by a nonzero vector and the coordinate axes.
(alpha, beta, gamma)
What is a direction vector?
It is a vector parallel to a line that is used to describe the direction, or orientation, of the line in space.
What is the result of a dot product and what is its usefulness?
It results in a scalar and it’s useful for determining if 2 vectors are orthogonal; if x * y = 0, then x & y are orthogonal.
It is used in projection calculations and W = F * s
What are equivalent vectors?
They are vectors that have the same magnitude and the same direction.
What are the 2 forms for the equation of a plane?
There is ax + by + cz + d = 0, where n = 〈 a, b, c 〉
is a normal vector of the plane, and (all vectors)
n * (x-p) = 0 where x = 〈 x, y, z 〉 and p is a point on the plane.
What are the parametric equations of a line?
They are the set of equations x, y, and z with the form = p + dt where d is the direction vector passing through point p.
What is the general equation for quadric surfaces?
The equation is Ax^2 + By^2 _ Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Jz + K = 0
What is the spherical coordinate system?
It is a way to describe a location in space with an ordered triple (ρ, θ, φ), where ρ is the
distance between P and the origin (ρ ≠ 0), θ is the same angle used to describe the location in cylindrical
coordinates, and φ is the angle formed by the positive z-axis and line segment OP, where O is the origin and
0 ≤ φ ≤ π
What’s the standard equation of a sphere?
It is (x − a)^2 + (y − b)^2 + (z − c)^2 = r^2 where the center is (a, b, c) and the radius is r
What are the symmetric equations of a line?
the equations are (x − x0)/a =
(y − y0)/b = (z − z0)/c where the direction vector is
v = 〈 a, b, c 〉 passing through point (x0, y0, z0)
What’s the three-dimensional rectangular coordinate system?
It is a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple (x, y, z) that plots its location relative to the defining axes
What’s a trace?
It is the intersection of a three-dimensional surface with a coordinate plane
