Math 126, Autumn 2021 Final Examination Page 1 of 10
1. (3 points per part) Suppose a and b are nonzero vectors in R3 . Decide whether each of the
following statements is always true, sometimes true, or never true. (Circle one.)
If your answer is always or never, briefly explain why (one sentence is enough).
If your answer is sometimes, give an example where it’s true and an example where it’s false.
(a) a · a > 0 0
Always Sometimes Never
Remember, for full credit, you must include a short explanation (for Always or Never) or examples (for Sometimes)!
2
a- a- •
=/ oil and / a- / =/ 0 because a- isn't Ñ .
,
(b) a ⇥ b = 2a Always Sometimes
0
Never
but
Never : a- ✗ b- is orthogonal to 5
,
2ñ can't be (if a- =/ 5)
ftp.0#eneverin0---cos0-)
(c) |a ⇥ b| = a · b Always Sometimes Never
Yes : a- =L ! !? b- =
No Whenever
: ⑦ =/ I a- 41,0 ? 5=42 :o)
4. e.g
, ,
.
(d) compa b > |b| Always Sometimes
0 Never
compete =/ b- / cos
-
0
never > I
(e) proja b = b Always
0 Sometimes Never
Yes : a- & b- parallel , e.g .
a- =( 1,903 b- 42,30>
,
other
No :
any time
, e.g . a- =
{ 1. 0,0} b- =p 1,0J
,
, Math 126, Autumn 2021 Final Examination Page 2 of 10
2. (4 points per part) Consider the vector function r(t) = h3 cos(t) + 1, 4 cos(t) + 2, 5 sin(t) + 7i.
(a) The space curve for r(t) lies in a plane. Find the equation of that plane.
4✗ = 12 cost 1-4
3y= 12 cost +6 product
find 3
pts and use cross .
or
,
I
4xt2
(b) Find parametric equations for the line tangent to r(t) at (1, 2, 2).
F' (f) =
f- 3
sing
-4cost 5- cost
,
> =
c-
F' (-2-1)=43 4,0> ,
✗ =/ +3T
y=2t4t
(c) Find T(t), the unit tangent vector to r(t).
/ Elt) / =fÉ+iÉt=5
Flt
)=¥¥-y=f¥si ¥si÷
1. (3 points per part) Suppose a and b are nonzero vectors in R3 . Decide whether each of the
following statements is always true, sometimes true, or never true. (Circle one.)
If your answer is always or never, briefly explain why (one sentence is enough).
If your answer is sometimes, give an example where it’s true and an example where it’s false.
(a) a · a > 0 0
Always Sometimes Never
Remember, for full credit, you must include a short explanation (for Always or Never) or examples (for Sometimes)!
2
a- a- •
=/ oil and / a- / =/ 0 because a- isn't Ñ .
,
(b) a ⇥ b = 2a Always Sometimes
0
Never
but
Never : a- ✗ b- is orthogonal to 5
,
2ñ can't be (if a- =/ 5)
ftp.0#eneverin0---cos0-)
(c) |a ⇥ b| = a · b Always Sometimes Never
Yes : a- =L ! !? b- =
No Whenever
: ⑦ =/ I a- 41,0 ? 5=42 :o)
4. e.g
, ,
.
(d) compa b > |b| Always Sometimes
0 Never
compete =/ b- / cos
-
0
never > I
(e) proja b = b Always
0 Sometimes Never
Yes : a- & b- parallel , e.g .
a- =( 1,903 b- 42,30>
,
other
No :
any time
, e.g . a- =
{ 1. 0,0} b- =p 1,0J
,
, Math 126, Autumn 2021 Final Examination Page 2 of 10
2. (4 points per part) Consider the vector function r(t) = h3 cos(t) + 1, 4 cos(t) + 2, 5 sin(t) + 7i.
(a) The space curve for r(t) lies in a plane. Find the equation of that plane.
4✗ = 12 cost 1-4
3y= 12 cost +6 product
find 3
pts and use cross .
or
,
I
4xt2
(b) Find parametric equations for the line tangent to r(t) at (1, 2, 2).
F' (f) =
f- 3
sing
-4cost 5- cost
,
> =
c-
F' (-2-1)=43 4,0> ,
✗ =/ +3T
y=2t4t
(c) Find T(t), the unit tangent vector to r(t).
/ Elt) / =fÉ+iÉt=5
Flt
)=¥¥-y=f¥si ¥si÷
