Vectors
Vector Practice Problems:
Odd-numbered problems from 3.1 - 3.21
Reminder: Scalars and Vectors
Vector:
Scalar:
A number (magnitude)
with a direction.
Just a number.
I have continually asked you, “which
way are the v and a vectors pointing?”
av
+x
Vectors
A car drives 50 miles east and 30 miles north. What is
the displacement of the car from its starting point?
North
+y
Displacement is
a vector
(net
change in
30
miles
position)
North
50 miles East
+x
East
Describing a vector
+y
+x
A vector is described *completely* by two quantities:
magnitude
(How long is the arrow?)
&
direction
(What direction is the arrow pointing?)
Magnitude and direction
North
+y
e: e
ud lin
nit this
g
Ma of
gth
len
θ
Direction:
angle from
reference point
(here, “θ degrees
North of East”)
30
miles
North
50 miles East
+x
East
Vector notation
+y
d
c
+x
This vector written down:
A A cd
And its magnitude…
|A| |A| |cd|
Vector “components”
+y
Ay
“Vector change in
y direction”
Ay
Ax
“Vector change in
x direction”
Ax
+x
Basic vector operations
Translating vectors
Vectors are defined by ONLY
magnitude and direction.
+y
=
These are all the
SAME vector!
+x
=
=
Basic vector operations
Multiplying by -1
–V, has an equal magnitude but
opposite direction to V.
+y
—
=
+x
In which case does
A.
B.
C.
D.
=⎻ ?
Q17
Basic vector operations
Geometrically adding vectors
+y
Two vectors with
the SAME UNITS
can be added.
[m/s]
[m/s]
+x
Basic vector operations
Geometrically adding vectors
+y
+
tail
=?
tip
+x
When adding geometrically,
always add tail to tip!
Basic vector operations
Geometrically adding vectors
+y
+
=
vector + vector = vector
+x
This is called the “triangle method of addition”
Basic vector operations
Geometrically adding vectors
+y
+
=
+
=
+x
It’s commutative!
It doesn’t matter which one you add first.
If you were to add these two
vectors, roughly what direction
would your result point?
A.
B.
C.
D.
E. None of the above
V1 + V2 = VR
Translate the vector and
always add tail to tip!
Q18
What is
+
A.
B.
C.
D.
=?
Q19
E. None of the above
Basic vector operations
Geometrically subtracting vectors
+y
-
= ?
+x
When adding/subtracting
geometrically, always add tail to tip!
Basic vector operations
Geometrically subtracting vectors
+y
-
=
+ (- )
+x
When adding/subtracting
geometrically, always add tail to tip!
Basic vector operations
Geometrically subtracting vectors
+y
-
=
+ (- )
+x
When adding/subtracting
geometrically, always add tail to tip!
Vectors
A car drives 50 miles east and 30 miles north. What is
the displacement of the car from its starting point?
North
+y
30
miles
North
50 miles East
+x
East
Vectors
A car drives 50 miles east and 30 miles north, then 20
miles south. What is the displacement of the car from
its starting point?
North
+y
20
miles
South
R
+x
East
Fig. 3.4 in your book
In graphical addition/subtraction, the arrows should
always follow on from one another, and the resultant
vector should always go from the starting point to
the destination point in your summed vector path.
Basic vector operations
Scalar multiplication
3
-3
Multiplying a vector A and a scalar (i.e. number) k
makes a vector, denoted by kA.
Intermission
Vector arithmetic: components
North
+y
What is D
(the magnitude of
30
miles
North
A.
B.
C.
D.
E.
Dy
50 miles East
Dx
)?
58 miles
80 miles
20 miles
0 miles
58 m/s
+x
East
Q20
Think about the Pythagorean Theorem
Vector arithmetic: components
A
Ay
Ax
A, Ay, and Ax here are the
MAGNITUDES of the vectors drawn
(they don’t have hats and are not bold).
Vector arithmetic: components
A
Ay
Ax
The magnitude of a vector
component is its final number
minus initial number!
Vector arithmetic: components
yf
A
Ay
xi
|Ay| = Ay
=
yf - yi
yi
xf
Ax
|Ax| = Ax = xf - xi
Vector arithmetic: components
North
+y
What is the magnitude of
the x component of D ?
D
70
mi
30
les
Dy
o
Dx
+x
East
Q21
A.
B.
C.
D.
E.
60.6 miles
35.0 miles
40.4 miles
0 miles
31 degrees
Think about SOH CAH TOA!
Using trigonometry,
you can find all vector components and
angles given just a bit of information!
Look at the triangles, and think about what
you can figure out based on available info.
Ultimate rule of vector math
Don’t fear the vector.
To study:
Practice drawing/graphing vector operations.
Get used to vector and magnitude notations.
Practice solving for x, y components.
Practice solving for θ.
Diandra kicks a soccer ball to a max height of 5.4
m at a 20° angle from the ground with a speed of
30 m/s. What is the x (horizontal) component of
the initial velocity of the soccer ball?
TO START:
Draw your vector right triangle.
What are the sides?
Compare your triangle with your neighbor.
CAREFUL! You can’t do vector
arithmetic combining displacement
(5.4m) with speed (30m/s)!
+y
30 m/s
5.4 m
20°
+x