market
Find a unit vector u in the direction of the vector from P1(1, 0, 1) to P2(3, 2, 0).
2/3i+2/3j-1/3k
find the angle between u = i - 2j - 2k and v = 6i + 3j + 2k.
arccos(-4/21)
Find u X v and v X u if u = 2i + j + k and v = -4i + 3j + k.
u X v =-2i - 6j + 10k
v X u = -(u X v) = 2i + 6j - 10k
find the point in which the line meets the plane.
x=1-t, y=3t, z=1+t; 2x-y+3z=6
(3/2, -3/2 , 1/2) is the point
Rectangle Find the measures of the angles between the diago- nals of the rectangle whose vertices are A = (1, 0), B = (0, 3), C = (3,4), and D = (4,1).
AC⋅BD=2(4)+4(−2)=0
so the angle measures are all 90
find the length of the indicated portion of the curve
r(t)=ti+(2/3)t^3/2k,0≤t≤3
52/3
Linear combination Let u=2i+j,v=i+j, and w= i - j. Find scalars a and b such that u = av + bw.
2i+j=a(i+j)+b(i−j)=(a+b)i+(a−b)j⇒a+b=2 and a−b=1⇒2a=3⇒a=23 and b=a−1=21
Find the volumes of the tetrahedra
A(1, -1, 0),B(0, 2, -2),C(-3, 0, 3),D(0, 4, 4)
44/3
Find the points in which the line x=1+2t,y=-1-t, z = 3t meets the coordinate planes.
(1, -1, 0 )
SKETCH THE GRAPH
y^2+z^2-x^2=1
SKETCH THE GRAPH
x=4-4y^2-z^2
нууу
SKETCH THE GRAPH
z=y^2-1
Equal-range firing angles
What two angles of elevation will enable a projectile to reach a target 16 km downrange on the same level as the gun if the projectile’s initial speed is 400 m/sec?
39.3 or 50.7
Distance along a line Show that if u is a unit vector, then the arc length parameter along the line r(t) = P + t u from the point P0(x0, y0, z0) where t = 0, is t itself.
t
Sketch the graph
r(t) = ti + (cost)j + (sint)k
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