The points A(4,5,10) B(2,3,4) and C(1,2,-1) are three vertices of a parallelogram ABCD. Find vector and Cartesian equations for the sides AB and BC and find the coordinates of D.

I need help please! Very much appreciated! thanks!!!!!!!

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- Mar 4th 2010, 11:35 AMYehiaSimple 3D Vectors question!
The points A(4,5,10) B(2,3,4) and C(1,2,-1) are three vertices of a parallelogram ABCD. Find vector and Cartesian equations for the sides AB and BC and find the coordinates of D.

I need help please! Very much appreciated! thanks!!!!!!! - Mar 4th 2010, 12:13 PMHallsofIvy
A vector from A to B is <2- 4, 3- 5, 4-10>= <-2, -2, -6>. Parametric equations for that line are x= 4-2t, y= 5- 2t, and z= 10- 6t with t= 0 at A and t= 1 at B. The vector equation is <4- 2t, 5- 2t, 10- 6t> or $\displaystyle (4- 2t)\vec{i}+ (5- 2t)\vec{j}+ (10- 6t)\vec{k}$.

Similarly, a vector from B to C is <1- 2, 2- 3, -1-4>= <-1, -1, -5>. Parametric equations are x= 2- t, y= 3- t, z= 4- 5t with t=0 at B and t= 1 at C.

To find the point D, start at A= (4, 5, 10) and add the vector from B to C:

(5, 5, 10)+ (-1, -1, -5)= (4, 4, 5).