# Thread: Determinant & vector geometry problem

1. ## Determinant & vector geometry problem

1.) Show that

det ( 1+x 2 3 4
1 2+x 3 4
1 2 3+x 4
1 2 3 4+x)

= x^3 (x+10)

It should be easy but I dont know why I cannot get the answer
Please help!

2.) Consider the triangle ABC formed by the points A(3,2,1), B(4,4,2) and C(6,1,0). Find the coordinates of point D on BC such that AD is perpendicular to BC. (ans: 1/17 (80 50 22)T)

THANKS A LOT!!

2. Originally Posted by coeyz
1.) Show that

det ( 1+x 2 3 4
1 2+x 3 4
1 2 3+x 4
1 2 3 4+x)

= x^3 (x+10)

It should be easy but I dont know why I cannot get the answer
Please help!
What you tried some elimination on the matrix?

3. 2.) Consider the triangle ABC formed by the points A(3,2,1), B(4,4,2) and C(6,1,0). Find the coordinates of point D on BC such that AD is perpendicular to BC. (ans: 1/17 (80 50 22)T)
Any point on BC has coordinates x= 4+ (6- 4)t= 4+ 2t, y= 4+ (1- 4)t= 4- 3t, and z= 2+ (0- 2)t= 2- 2t.

Let D have coordinates (x, y, z). Then the vector AD is (x- 3)i+ (y- 2)j+ (z- 1)k and BC is 2i- 3j- 2k. The dot product of those two vectors is 0. Replacing x, y, and z with there expression in t gives a single equation for t.