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Question 1: The equation of a line can be determined using two points on the line. a) Find the vector, parametric, and symmetric equations of the...

Question 1: The equation of a line can be determined using two points on the line.

a) Find the vector, parametric, and symmetric equations of the line through the points (-2, 3, 1) and (1, 4, -2).

b) Explain the features of the equations of a line that is parallel to the xz plane, but does not lie on the plane, and is not parallel to any of the axes.

Question 2: Two given lines are either parallel, skew, or intersecting.

a) Determine, if there is one, the point of intersection of the lines given by the equations

(x-1)/(-3)=(y-8)/6=(z-3)/(-2) and (x-8)/4=(y+3)/(-5)=(z-7)/2

b) Given the equations of two lines that meet at the point (-1, 5, 2) and which meet at right angles, but do not use that point in either of the equations. Explain your reasoning.

Question 3: The equation of a plane can be determined using three points on the plane.

a) Find the vector, parametric, and general equations of the plane through the points (2, -3, 1), (3, 1, 6), and (5, -1, 2).

b) Given the equation of a plane that crosses the axes at points equidistant from the origin. Explain your reasoning.

Question 4: A line can either lie on a plane, lie parallel to it or intersect it.

a) Determine, if there is one, the point of intersection between the line given by the equation (x-7)/2=(y-9)/4=(z-4)/3 and the plane given by the equation

[x,y,z]=[5,4,-1]+s[2,3,1]+t[4,-1,-2]

b) Determine the angle between the line and the plane.

c) Give the equation of a plane and three lines, one of which is parallel to the plane, one which lies on the plane, and one of which intersects the plane. Explain your reasoning.

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