Math 2210
Practice Midterm #1 Solutions
Instructor: Mike Shrieve
Name:
The midterm is 50 minutes. No calculators or equation sheets are allowed. Answer all
questions.
(1) Let ~u = h4, 2, −7i and ~v = h−1, 3, 6i. Calculate the following:
(a) ~u + ~v
~u + ~v = h4 − 1, 2 + 3, −7 + 6i = h3, 5, −1i
(b) ~u · ~v
~u · ~v = (4 · −1) + (2 · 3) + (−7 · 6) = −40
(c) ~u × ~v
~u × ~v = (12 + 21)ı̂ − (24 − 7)̂ + (12 + 2)k̂ = h33, −17, 14i
(d) Proj~v ~u
Proj~v ~u =
~u · ~v
k~v k2
~v = −
20
40
h−1, 3, 6i = − h−1, 3, 6i = h 20
, − 60
, − 120
i.
23
23
23
46
23
(2) Write an equation for the plane passing through the points (1, 4, 5), (−2, 6, −1), and
(0, 0, 0).
Find two vectors parallel to the plane: ~v1 = h1, 4, 5i, ~v2 = h−2, 6, −1i. Compute
the cross product: ~v1 ×~v2 = (−4 − 30)ı̂ − (−1 + 10)̂ + (6 + 8)k̂ = h−34, −9, 14i. Now,
the equation for the plane is −34x − 9y + 14z = D. Plug in a point to find D = 0.
The equation is then −34x − 9y + 14z = 0.
(3) Write an equation for the plane passing through the origin which is orthogonal to
both the plane x + y + z = 7 and the plane 4x − 2y = 1.
Find the normal vectors of the two planes: ~n1 = h1, 1, 1i and ~n2 = h4, −2, 0i. Now
compute the cross product: ~n1 × ~n2 = (0 + 2)ı̂ − (0 − 4)̂ + (−2 − 4)k̂ = h2, 4, −6i.
So the equation for the plane is 2x + 4y − 6z = D. Plug in a point to find D = 0. So
the equation for the plane is 2x + 4y − 6z = 0.
(4) Write a parametric equation for the tangent line to the curve given by the vector
function ~r(t) = ht2 , 7t, 4t − 1i at t = 4.
2
To write the parametric equation for a line we need one point and the direction
vector. The point is ~r(4) = (16, 28, 15). And the direction vector is obtained by computing r~0 (t) = h2t, 7, 4i and evaluating at t = 4: r~0 (4) = h8, 7, 4i. Now the equation
for the line is (16 + 8t, 28 + 7t, 15 + 4t). In class I used s to parametrize the line to
emphasize that the line is parametrized separately from the curve itself. On the test
it is fine to use either, but you should keep in mind that the point (16, 28, 15) is on
the curve at t = 4 and on the line at t = 0.