B04. 136.130: Test #4 Solutions

1. Suppose and . Find

(a)

(b)

Solution.

(a)

(b) .

2. We are given a point and a vector .

(a) Find the point normal equation of the plane through P and orthogonal to u.

(b) Find the parametric equations of the line through P and parallel to u.

Solution.

(a)

(b)

3. The plane is given by the standard equation , and the line l is given by the parametric equations .

(a) Find the coordinates of the intersection point of and l.

(b) Is the line l orthogonal to the plane? Justify your answer.

Solution.

(a) Substitute in the equation of to get . Solve for t to get . Put this in to get . So, the intersection point is (0,3,1).

(b) The vector (1,-1,2) is orthogonal to the plane . The vector (-1,1,-2) is parallel to l. Since the former is (-1) times the latter, these two vectors are parallel. So, the line l is orthogonal to the plane .