the line "l" passes through the point P=(1,-1,1) and has a direction vector d=[2,3,-1], for the following plane Pl, determine whether l and P is parallel, perpendicular, or neither.
a) 2x + 3y - z = 1
could you tell me also how it would differ if it was parallel or neither or perpendicular?
Thank You!!
To determine whether the line "l" and the plane "Pl" are parallel, perpendicular, or neither, we can compare their direction vectors.
First, let's find the normal vector of the plane "Pl". The coefficients of x, y, and z in the equation of the plane (2x + 3y - z = 1) give us the normal vector [2, 3, -1].
Next, compare the direction vector of the line "l" ([2, 3, -1]) with the normal vector of the plane.
If the direction vector and the normal vector are parallel, then the line and plane are parallel.
If the dot product of the direction vector and the normal vector is zero, then the line and plane are perpendicular.
Otherwise, if the dot product is neither zero nor parallel, then the line and plane are neither parallel nor perpendicular.
To calculate the dot product, multiply the corresponding components of the two vectors and sum them up.
For the given line "l" with direction vector d=[2, 3, -1], and the plane "Pl" with normal vector [2, 3, -1]:
Dot Product = (2 * 2) + (3 * 3) + (-1 * -1) = 4 + 9 + 1 = 14.
Since the dot product is not zero, line "l" and plane "Pl" are not perpendicular.
Since the dot product is not parallel (not zero), line "l" and plane "Pl" are not parallel either.
Therefore, line "l" and plane "Pl" are neither parallel nor perpendicular.