Austin Cholley Zhu MAT 267 ONLINE A Fall 2021
Assignment Exam 1 due 09/03/2021 at 08:23pm MST
Problem 1. (1 point)
Find the distance from the point (3, -4, 4) to the plane −x+4y+
5 2.z=
Answer(s) submitted:
•.154303
(correct)
Correct Answers:
•0.154303349962092
Problem 2. (1 point)
Find the area of the triangle with vertices:
Q( ) ( )2,4 3, , R1,1 1, , S( )4 1, , 2.
Solution:
SOLUTION:
Let a=~
QR =h− − −1,3,2iand b=~
QS =h − − i2,3,1 . The area
of the triangle is equal to half the lenght of the cross product of
these two vectors.
The corss product is given by , and it has mag-a×b=h−3,−5 9,i
nitude | −a×b|=p(3)2+ (−5 115.)2+ (9)2=√
Therefore the area of the triangle is 1
2√115.
Answer(s) submitted:
•5.362
(correct)
Correct Answers:
•5.3619026473818
Problem 3. (1 point)
Gandalf the Grey started in the Forest of Mirkwood at a point P
with coordinates and arrived in the Iron Hills at the point(−2,−2)
Qwith coordinates (0, 2). If he began walking in the direction of
the vector v=5i +2j and changes direction only once, when he
turns at a right angle, what are the coordinates of the point where
he makes the turn?
( , )
Solution:
SOLUTION:
Let w=~
PQ =h2,4iand let p=v·w
| |v2v=18
29 h5 2,i=h90
29 ,36
29 ibe the
vector projection of wonto .v
The point at which Gandalf the Grey changes direction is given by
(−2+90
29 ,−2+36
29 ) = ( 32
29 ,−22
29 )
Answer(s) submitted:
•-2-((400)/(29))
•-2-((160)/(29))
(incorrect)
Correct Answers:
•1.10344827586207
•-0.758620689655172
1
Problem 4. (1 point)
Find the solution of the differential equation with the givenr(t)
initial condition:
r0(t) = hsin 3t t,sin 4 ,9ti, , ,r(0) = h4 9 4i
r(t) = h, , i
Solution:
Solution: We first integrate the vector to find the generalr0(t)
solution:
r(t) = Zhsin 3t 4t 9t dt,sin ,i
=Zsin 3tdt tdt,Zsin 4 ,Z9tdt=−1
3cos3t,−1
4cos4t,9
2t2+c
Substituting the initial condition we obtain:
r(0) = −1
3cos0,−1
4cos0,9
202+c
=h4 4,9,i=−1
3,−1
4,0+c
Hence,
c=h4,9 4,i−−1
3,−1
4,0=13
3,37
4,4
