Give the ranks of both the matrix of coefficients and the augmented

[salman213]

Hey!
I was looking at some problems since I have a mid term tomorrow and don't get how to do this one perhaps someone can help!


http://img143.imageshack.us/img143/3452/16856739ys6.jpg


I don't know if I am doing this right but i went backwards and found the vector equation to look like

x = 1 + 3t - s
y = s
z = t

(x,y,z) = (1,0,0) + t(3,0,1) + s(-1,1,0)
[1 1 3 l 1]
[0 0 0 l 0]
[0 0 0 l 0]


do the rank of the augmented matrix is 1 and does that mean the rank of the coefficient matrix is also 1 ?

b) for b part should i just plug in that coordinate in the x,y,z vecotr form to solve for s, and t and hope for the same answer to show it is a solution

c) for c should i use the two vectors given from the vector equation s(-1,1,0) and t(3,0,1)

d) not sure how to find distance between two planes :S


CAN SOMEONE TELL ME IF IM COMPLETELY LOST?
and cross them to get the normal then use the point given to get the scalar equation

 

[salman213]

ok you can delete this thread now, my midterm already happened so its kinda useless now..lol thanks anyways

 

[tacman]

First of all, it seems like you are on the right track with finding the vector equation for the system of equations. However, the ranks of the coefficient matrix and the augmented matrix are not the same in this case. The rank of the coefficient matrix is 2, while the rank of the augmented matrix is 1. This means that there is one independent variable in the system.

To solve for the coordinates in the vector equation, you can plug in the given point (1,0,0) into the equation and solve for t and s. This will give you the values for t and s that satisfy the system of equations.

For part c, you can use the two vectors given in the vector equation, (3,0,1) and (-1,1,0), to find the direction vectors of the two planes. Then, you can use the cross product to find the normal vector and use the given point to find the scalar equation of the planes.

For part d, to find the distance between two planes, you can use the distance formula that involves the normal vectors and the distance between the planes. You can also use the point-to-plane distance formula, where you use the given point and the normal vector to find the distance between the point and the plane. I suggest looking up these formulas and trying them out to see which one works best for you.

Overall, it seems like you have a good understanding of the concepts, but you may need to review the formulas and methods for solving these types of problems. Good luck on your midterm!