PreCalculus Formulas for Sections 9-5 to 9-6
Magnitude of vector:
a 2  b2
||v|| =
Unit vector:
u=
v
v
Vector in terms of magnitude and direction:
v = v (cos  i  sin  j)
Dot product:
If v = a1i + b1j and w = a2i + b2j
then, vw = a1a2 + b1b2
Angle between vectors:
vw
v w
cos  =
Decomposition of vectors where v1 is parallel to w and v2 is orthogonal to w:
v1 =
vw
w
2
v2 = v – v1
w
Work:
W = Fhorizontal distance
Directions angles:
cos  
a
a b  c
2
2
2
cos  
b
a b  c
2
2
2
cos  
Writing a vector in terms of its magnitude and direction cosines:
v || v || cos i  cos j  cos  k 
c
a  b 2  c2
2
PreCalculus 2 Sample Quiz2 9.5-9.6
Name________________________________________
1. Find the dot product v · w given the two vector. Find the angle between v
and w and state whether the vectors are parallel, orthogonal, or neither.
v = 3i +4j, w = -6i -8j
2. Find the work done by a force of 4 pounds acting in the direction of 60° to the
horizontal in moving an object 8 feet from (0,0) to (8, 0)
3. A car with a gross weight of 5300 pounds is parked on a street with a slope of
8°. Find the force required to keep the car from rolling down the hill.
Weight  sin  What is the force perpendicular to the hill? Weight  cos 
4. Find the distance from P1 = (4, 3, -2) to P2 = (-5, -6, 7).
5. A vector v has initial point P = (1, 3, -2) and terminal point (5, -3, 2). Write v
in the form v = ai + bj + ck.
6. Given v = 5i +4 j – 2k and w = -3i + 2j – 5k find v  w .
7. Find the unit vector for v  3i  6 j  2k
8. Find the dot product and the angle between v and w given
v = 4i – j + 2k and
w = i – 2j – 4k
9. Find the direction angles of the vector v = 2i – 4j + 4k.
10. Find the equation of a sphere given the radius and the center
r  4 P  (3 , 2,  1)
11. Find the radius and center of the sphere
x2  y 2  z 2  2x  6 y  2
12. A plane has an airspeed of 700 km/hr bearing S 30 E . The wind velocity is
30km/hr in the direction S 60 E . Find the resultant vector representing the
path of the plane relative to the ground. What is the ground speed of the
plane? What is its direction?