If a problem is solved. It is not 'the' answer.
No attempt is made to search for the most elegant answer.
I highly recommend that you at least try to solve the
problem before you read the solution.
Take line AB with A(4,5,6) and B(6,7,8). Give direction numbers of that line. Is C(1,2,3) on that line? |
Write the parametric equations and cartesian equations of the x-axis. |
Take the triangle ABC with A(2,2,4) B(4,6,0) and C(0,0,2). Calculate the median lines. |
Calculate the parametric equations and cartesian equation of the plane formed by the x-axis and the y-axis. |
Calculate the cartesian equation of the
plane containing the point A(1,2,3) and parallel to the lines b and c
b: 4x = 3y ; z = 2 and c: -5x + 3y + 2=0 ; x + z = 4 |
The plane ABC has equation 4x - 3y - z + 5 = 0. Calculate the equation of the plane parallel to ABC and containing point D(2,1,3). |
Given : x - 4 y - 6 z - 2 line b: ------ = -------- = ------ -3 -1 3 x - 1 y - 2 z - 3 line c: ------ = -------- = ------ -1 -2 2 Calculate the equation of the plane such that A(1,2,3) is in that plane and that b and c are parallel to that plane. |
Given : x - 4 y - 6 z - 2 line b: ------ = -------- = ------ -3 -1 3 x - 1 y - 2 z - 3 line c: ------ = -------- = ------ -1 -2 2 Are these lines orthogonal? |
Take plane ABC: 3x-2y-4z=3 and plane DEF: x-y-z=3. Are these planes orthogonal? |
The points P(2,-2,1) and Q(1,2,-2) belong to a sphere with center O(0,0,0). Calculate the angle between the two tangent planes in P and Q to the sphere. |
Are the lines b and c intersecting? parallel? / x = 4 + r.(-3) b: | y = 6 + r.(-5) \ z = 0 + r.3 / x = 3 + r.3 c: | y = 1 + r.1 \ z = 1 + r.3 |
Are the lines b and c intersecting? parallel? line b: 2x + 3y + z = 6 ; x + y + z = 3 line c: x + 2y - z = 2 ; x - z = 0 |
Calculate the orthogonal projection A' of point A(1,2,3) on the plane 3x-y+4z = 0. |
Calculate the sharp angle between the lines / x = 1 + r | y = 2 - r \ z = 1 + r and / x = 1 + r.3 | y = 2 \ z = 3 + r.4 |
Calculate the sharp angle between the planes 2x + y + 4z = 2 and x + y - 4 = 0 |
Given: A(2,1,0) ; B(1,0,1) ; C(3,0,1) D(0,0,2) Point D is on a line l orthogonal to the plane ABC. Calculate the equations of l, the intersection point S with the plane and the distance from D to the plane ABC. |
Find the equation of the plane which is perpendicular bisector of the segment [AB] with A(1,2,3) and B(5,6,7). |
Take a plane x + y - z = 1 and point A(1,2,-3). A line l has equations / x = 1 + r.3 | y = 2 + r.(-1) \ z = 3 + r.4 Calculate the coordinates of a point B of line l, such that AB is parallel to the plane. |
Take a point A(1,2,0).
A line l has equations
/ x = 1 + r | y = 2 - r \ z = 1 + rCalculate the coordinates of the points B of line l, such that |AB| = sqrt(6). |
Two planes have respectively an equation
2 x - 2 y - z + 5 = 0 and x + 5 y - z - 8 = 0Point A = A(3,5,7) Find the equation of a plane through A and perpendicular to the two planes. |
Given: The plane alpha with equation 2x + 3y - z -7 = 0 The line d with equations [ 3x + y - z = 0 ; x - y - z + 2 = 0 ] Find the plane gamma through d ans perpendicular to alpha. |
A plane alpha has an equation x + y + z = 3. P(1,1,1) is in alpha. Point P is on the line d and d has a direction vector (1,2,3). Find the line c, in alpha, such that c and d are orthogonal lines and P is on c. |
A pyramid has base ABC
A(1, 2, -3) ; B(0, -1, 5) ; C(-3, 0, 9)The vertex T is a variable point of the line x = 1 + r y = 2 + r z = -1 - 2 r1) Find T such that the height of the pyramid is 4. 2) Find the volume of the pyramid. |
M is the centroid of the triangle DEF with D(6,0,0), E(0,6,0) and F(0,0,6). C is a circle, in plane DEF, with radius 2 en center M. Find point L on C such that L is as close as possible to the plane z = 0. |
Given: a fixed point P and two variable lines a and b. Find the line c through point P such that line c intersects both lines. Find the value of the parameter so that there are infinitely many solutions P(3,1,6) line a / x + (m-1)y -2 = 0 \ y + z - 3 = 0 line b / (2m-3)x - 1 = 0 \ y - z + 1 = 0 |