Exam 11: Vectors and the Geometry of Space

Find parametric equations of the line through parallel to .

Compute ||-2a - 4b||.

Use vectors to determine if the points (3, -1, 0), (3, -3, -1), (0, 1, -9), and (0, 3, -8) form a square.

Identify the plane as parallel to the xy-plane, xz-plane, or yz-plane. y = -3

Find symmetric equations of the line through and normal to the plane .

Suppose that there are two forces acting on a barge being towed along a river. One force is exerted due south by the river current and has a magnitude of 7 units (the exact nature of the force unit is unimportant). The other force is exerted by a tugboat and has a magnitude of 11 units towards the north and 3 units towards the east. What is the net force acting on the barge?

Find a vector perpendicular to the given vector. Show all your work.

Determine if the lines are parallel, skew or intersect.

Match the equation to its graph. [The figures below are not necessarily drawn to scale.]

Match the equation to its graph. [The figures below are not necessarily drawn to scale.]

In many farmers' barns and in many mechanics' shops, you can find a stout steel pipe about 2 to 3 feet in length with one end pinched so that the pinched end fits snugly over a wrench handle. What might explain the common presence of this object in these settings? Explain your answer in terms of the vector concepts you have learned.

Find parametric equations of the line through and .

Find the displacement vectors and and determine whether the points P = (-1, -5, 2), Q = (1, -3, 4), and R = (0, -4, 3) are collinear (on the same line).

Suppose two airplanes fly paths described by the parametric equations where s and t represent time. Determine whether the flight paths intersect and, if so, whether the planes collide.

Which one of the following statements is always true?

Identify the surface.

Determine if the vectors are orthogonal.

Determine if the lines are parallel, skew or intersect.

Determine whether the given lines are the same. x = -7 + 4t, y = 1 + t, z = -2 - 7t and x = -3 + 8t, y = 2 + 2t, z = -9 - 14t

Determine whether the given planes are the same. -3(x + 4) + 2(y - 6) - (z + 5) = 0 -3(x - 8) + 2(y + 8) - (z + 69) = 0