B08. 136.130: Test #4 Solutions

1. Suppose and . Find

(a) The cosine of the angle between u and v.

(b)

Solution.

(a)

(b) .

2. We are given points and .

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

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

Solution.

=(0,4,2).

(a)

(b)

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

(a) Find the coordinates of the intersection point of m and l (there is exactly one intersection point).

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

Solution.

(a) It is visible form the equations that the point (1,2,3) lies on both lines. Alternatively, solve all of the equations to get it (t will be 0).

(b) The vector (-2,1,1) is parallel to the line m (visible from the equation). By the same argument, the vector (-1,1,-3) is parallel to l. Since the dot product of these two vectors is 2+1-3=0, it follows that they are orthogonal. So, the lines are orthogonal too.