calculus please help

Question

Find a nonzero vector parallel to the line of intersection of the two planes 4𝑧−(4𝑥+𝑦)=54z−(4x+y)=5 and 3𝑧−(3𝑥+4𝑦)=−13z−(3x+4y)=−1. 

Answer 1

We first solve for the line of intersection between these two planes. To do this, we know that 4z-(4x+y)=5 and 3z-(3x+4y)=-1. At the intersection, both equations must hold. Solving the first equation for y, we see that y=4z-4x-5. Plugging this into the second equation, we see that 3z-(3x+4(4z-4x-5))=-1, so simplifying, 13x-13z=-21. Therefore, since y=4z-4x-5, since z-x=21/13, we see that y=84/13-5=19/13 for all points on the intersection. This describes the intersection, z=x+21/13 and y=19/13. To find the vector parallel to this, we simply need to move the line so that it passes through the origin, so this would be parallel to the line z=x and y=0, which is represented by the vector (1,0,1).