CK-12-Pre-Calculus Concepts

7.3. Resolution of Vectors into Components http://www.ck12.org

3. The vector−→BPis<− 7 , 13 >. First take the unit vector and then scale it so it has a magnitude of 10.

BP

|BP|=<

√−^7

218 ,

√^13

218 >

10 ·|BPBP|=< √− 21870 ,√^130218 >

You end up with a vector that is ten units long in the right direction. The question asked for a point fromBwhich
means you need to add this vector to pointB.
( 4 ,− 5 )+<√− 21870 ,√^130218 >≈(− 0. 74 , 3. 8 )

Practice

−→v =< 1 ,− 3 >,−→u =< 2 , 5 >,−→t =< 9 ,− 1 >,−→r =< 2 ,y>

A= (− 3 , 2 ),B= ( 5 ,− 2 )

1. Solve foryin vector−→r to make−→r perpendicular to−→t.


2. Find the unit vector in the same direction as−→u.


3. Find the unit vector in the same direction as−→t.


4. Find the unit vector in the same direction as−→v.


5. Find the unit vector in the same direction as−→r.


6. Find the point exactly 3 units away fromAin the direction ofB.


7. Find the point exactly 6 units away fromBin the direction ofA.


8. Find the point exactly 5 units away fromAin the direction ofB.


9. Jack and Jill went up a hill to fetch a pail of water. When they got to the top of the hill, they were very thirsty so
   they each pulled on the bucket. Jill pulled at 30◦with 20 lbs of force. Jack pulled at 45◦with 28 lbs of force. What
   is the resulting vector for the bucket?


10. A plane is flying on a bearing of 60◦at 400 mph. Find the component form of the velocity of the plane. What
    does the component form tell you?


11. A baseball is thrown at a 70◦angle with the horizontal with an initial speed of 30 mph. Find the component
    form of the initial velocity.


12. A plane is flying on a bearing of 200◦at 450 mph. Find the component form of the velocity of the plane.


13. A plane is flying on a bearing of 260◦at 430 mph. At the same time, there is a wind blowing at a bearing
    of 30◦at 60 mph. What is the component form of the velocity of the plane?


14. Use the information from the previous problem to find the actual ground speed and direction of the plane.


15. Wind is blowing at a magnitude of 40 mph with an angle of 25◦with respect to the east. What is the velocity of
    the wind blowing to the north? What is the velocity of the wind blowing to the east?