1. Use normal vectors to determine the interaction, if any, for each of the following pairs of planes. Give a geometric interpretation in each case and the number of solutions for the corresponding system of linear equations. If the planes intersect in a line, determine a vector equation of the line.
a. x + 2y + 3z = ?4
2x + 4y + 6z = 10
2x ? y + 2z = ?8
b. 4x ? 2y + 4z = ?16
x + 3y ? 5z = ?12
2x + 3y ? 4z = ?6
2. Use normal vectors to determine the intersection, if any, for each of the following groups of three planes. Give a geometric interpretation in each case and the number of solutions for the corresponding linear system of equations. If the planes intersect in a line, determine a vector equation of the line. If the planes intersect in a point, determine the coordinates of the point.
a. x ?2y ?2z = 6
2x ? 5y + 3z = ?10
3x ? 4y + z = ?1
b. x ? y + 3z = 4
x + y + 2z = 2
3x + y + 7z = 9
any intersection of two planes results in a straight line
like we say that point of intersection of two lines
it is line of intersection of two planes.
so when you are given three planes, there would be three lines of intersections (taking two planes at time)
maybe it will gt easier for you to solve when you do it with three lines.
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