Algebra & Trig, Sullivan & Sullivan
Fourth Edition
Notes:§9.4
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§9.4 – Vectors
I.
Introduction to vectors
A.
What is a vector?
1.
2.
II.
Suppose you walked 3 miles east then 4 miles north from your home.
a.
How far did you travel?
b.
How far are you from your home?
c.
What is the angle from your home to your current location?
Give a definition of vector.
a.
What does the length of the vector indicate?
b.
What does the orientation of the vector indicate?
3.
Discuss what is meant by ‘adding vectors’. Show that the sum of two vectors gives net
displacement.
4.
Example: A person leaves her home and walks 5 miles due east and then 3 miles
northeast.
a.
How far has she walked?
b.
How far away from home is she?
c.
What is her net displacement?
Decomposition of vectors
A.
Suppose a vector is situated so that its initial point, P, and terminal point, Q, are as given. Find the
components of the vector and write in <a, b> form.
1.
P = (0, 0); Q = (3, 4)
2.
P = (1, 5); Q = (8, 10)
3.
P = (-2, -6); Q = (5, -1)
4.
What does it mean for two vectors written in component form to be equal?
Algebra & Trig, Sullivan & Sullivan
Fourth Edition
B.
C.
C.
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Basic operations on vectors, graphically
1.
How do you add two vectors graphically?
2.
What does it mean to multiply by -1?
3.
What does it mean to multiply by any other scalar? Zero?
4.
How do you subtract two vectors graphically?
5.
What does it mean that two vectors are equal?
Unit vectors
1.
D.
Notes:§9.4
Define the vectors i and j.
Basic operations on vectors,component-wise
1.
Addition is just adding each part (the x part & y part), independent of each other)
2.
SCALAR multiplication involves multiplying each part of the component by the same
number (i.e., doubling the distance traveled in each direction)
3.
How do you add, subtract, and scalar multiply vectors in component form?
a.
v  3i  4 j and w  2i  5 j , find v  w , 2v , and 2v  w
b.
v  4i  5 j and w  i  3 j , find v  w , 3w , and v  3w
Magnitude / direction
1.
How do you find the magnitude and direction of a vector in component form?
2.
First put it in the format of an algebraic vector, then
Magnitude: use the distance formula to figure this out
Direction: Use tan-1, making sure you've got the quadrant correct.
2.
Examples
a.
v  3i  4 j ; find v and 
b.
w  24i  7 j ; find w and 
Remember that the goal is to put the vector into the form
v  v cos i  sin j 
(where v is a vector, and i & j are the unit vectors)
D.
Composition of a vector given magnitude and direction
1.
What formula can you use to find the components of a vector given its magnitude and
direction?
2.
v  v cos i  sin j  , then distribute out
Write the vector in the form ai + bj, given its magnitude and direction (angle from positive
x-axis).
a.
v  5 ,   60
b.
w  7 ,   330
Algebra & Trig, Sullivan & Sullivan
Notes:§9.4
Page 3 / 3
Fourth Edition
3.
Using the vectors above, find v  w . Why can’t you just add the magnitudes and angles?
III.
Application – resultant force
Two forces act on an object at angles of 30° and -45° as shown in the figure.
Find the direction and magnitude of the resultant force.
C.
E.
Unit vectors
1.
Define the vectors i and j.
2.
How do you convert a position vector into unit vector form?
3.
How do you add, subtract, and scalar multiply vectors in component form?
a.
v  3i  4 j and w  2i  5 j , find v  w , 2v , and 2v  w
b.
v  4i  5 j and w  i  3 j , find v  w , 3w , and v  3w
Normalizing a vector
1.
What does it mean to normalize a vector?
2.
Find the unit vector having the same direction as v.
a.
v  3i  4 j
b.
v  24i  7 j