Vector operations
Equations
Planes
Lines
Angles and distances
100
What is the sum of vectors [-1, 2, 4] and [1, 0, -1
][0, 2, 3]
100
Find the parametric equation of the line through A(2, -1, 4) and B(-1, 0, 2)
[x, y, z] = [2, -1, 4] + t [-3, 1, -2]
100
What is the normal vector of the plane that is parallel with vectors [2, 2, -2] and [-2, 3, -5]?
any non-zero multiple of [2, -7, -5]
100
State the set of parametric equations of the line that passes through (1, -2, 3) and is parallel with [4, 5, -6].
(x, y, z) = (1 +4t, -2 + 5t, 3 - 6t)
100
What is the distance between (-1, 3, 2) and (2, 1, -4)
7
200
What are the coordinates of the midpoint of (-1, 2, 40) and (1, 0, -1)?
(0, 1, 1.5)
200
Find the angle between x + 2y - z = 8 and (x, y, z) = (t, 1 - t, 3 + 2t).
30 degrees
200
Find the coordinates of the foot of the normal from (A2, -1, 3) to the plane x - y + 2z = 27.
(5, -4, 9)
200
Find the angle between the lines (x, y, z) = (2 - 3t, -1 + t, 4 - 2t) and (1 - x)/3 = y = (2 - z)/2
0 degrees
200
What is the area of the triangle with vertices A(-1, 2, 3), B(2, 1, 4) and C(0, 5, -1)?
0.5 sq. rt. (270)
300
In a parallelogram, A(-1, 2, 1), B(2, 0, -1) and D(3, 1, 4). What are the coordinates of C?
C(6, -1, 2)
300
Find a vector parallel to (x - 1) / 2 = (3 - y) / 3 = z.
Any non-zero multiple of (7, -3, 2).
300
Determine the coordinate of the point where the line through A(-1, 2, 3) and B(2, 0, -3) intersects the plane x - 2y + 3z = 26.
(-7, 6, 15)
300
Find the angle between (x, y, z) = (2, -1, 3) + t (-1, 2, 1) and (x - 1) / 2 = (3 - y) / 3 = z.
40.2 degrees
300
What is the angle between x + y - z = 8 and 2x - y + 3z = -1?
72.0 degrees
400
What is the cross product if [1, 2, -1] and [1, 0, -3]?
[-6, 2, -2]
400
Find the coordinates of the foot of the perpendicular from P(-1, 2, 3) to (x, y, z) = (1 + 2t, -4 + 3t, 3 + t).
(3, -1, 4)
400
Find the shortest distance from (2, -1, 1) to 3x - 2y + 7z = 6.
9 / sq. rt. 62
400
Find the equation of the line that is perpendicular to (x - 1) / 2 = (3 - y) / 3 = z and passes through (5, -3, 2).
[x, y, z] = [5, -3, 2] + s [0, 1, 3]
400
Find the distance from (2, -1, 3) to (x - 1) / 2 = (3 + y) / 3 = z.
3.73 units
500
Find the value of x for which the points A(1, 1, 2), B(2, 4, 0), C(3, 1, 1) and D(x, 0, 1) are coplanar.
4
500
Given the planes 2x + 4y + z = 1, 3x + 5y + 1 and 5x + 13y + 7z = 4, find the coordinates of all points that lie on these three planes.
(0 + t, 0.2 - 0.6t, 0.2 + 0.4t)
500
Find the coordinates of the foot of the perpendicular from (2, -1, 3) to [x, y, z] = [1, 0, 3] + t [4, -1, 1] + s [2, 1, -2].
(2 + 9/137, -47/137, 3 + 54/137)
500
What is the shortest distance between (x, y, z) = (t, 1 - t, 2 + t) and (x, y, z) = (3 - s, -1 + 2s, 4 - s)?
1 / sq. rt. 2
500
What is the volume of the tetrahedron with vertices P(0, 0, 1), Q(2, 3, 0), R(-1, 2, 1) and S(1, -2, 4)?
3.5
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