08 ch 3a vectors - home | sarah spolaor · vectors vector practice problems: odd-numbered problems...
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VectorsVector Practice Problems:
Odd-numbered problems from 3.1 - 3.21
Reminder: Scalars and VectorsVector:
A number (magnitude) with a direction.
a +xv
I have continually asked you, “which way are the v and a vectors pointing?”
Scalar:Just a number.
Vectors
North +y
+x East
A car drives 50 miles east and 30 miles north. What is the displacement of the car from its starting point?
Displacement is a vector
(net change in position)
50 miles East
30 miles North
A vector is described *completely* by two quantities:magnitude
(How long is the arrow?)
direction(What direction is the arrow pointing?)
Describing a vector+y
+x
&
Magnitude and direction
North +y
+x East
50 miles East
30 miles North
Magnitude:
length of this lin
e Direction:angle from
reference point (here, “θ degrees
North of East”)θ
Vector notation+y
+x
c
d
This vector written down: cdAA
And its magnitude…|cd||A||A|
Vector “components”+y
+x
Ax
Ay
Ax
Ay
“Vector change in y direction”
“Vector change in x direction”
Basic vector operations
+y
+x
Vectors are defined by ONLY magnitude and direction.
= = =
These are all the SAME vector!
Translating vectors
Basic vector operations
+y
+x
–V, has an equal magnitude but opposite direction to V.
— =
Multiplying by -1
In which case does = ⎻ ?A. B.
C. D.
Q17
Basic vector operations
+y
+x
Two vectors with the SAME UNITS
can be added.
[m/s]
[m/s]
Geometrically adding vectors
Basic vector operations
+y
+x
+ = ?
Geometrically adding vectors
When adding geometrically, always add tail to tip!
tiptail
Basic vector operationsGeometrically adding vectors
+y
+x
+ = vector + vector = vector
This is called the “triangle method of addition”
Basic vector operationsGeometrically 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? Q18
E. None of the above
A. B.
C. D.
Translate the vector and always add tail to tip!
V1 + V2 = VR
What is Q19
E. None of the above
A. B.
C. D.
+ = ?
Basic vector operations
+y
+x
Geometrically subtracting vectors
When adding/subtracting geometrically, always add tail to tip!
- = ?
-
Basic vector operations
+y
+x
Geometrically subtracting vectors
- = + (- )
When adding/subtracting geometrically, always add tail to tip!
-
Basic vector operations
+y
+x
Geometrically subtracting vectors
- = + (- )
When adding/subtracting geometrically, always add tail to tip!
Vectors
North +y
+x East
A car drives 50 miles east and 30 miles north. What is the displacement of the car from its starting point?
50 miles East
30 miles North
Vectors
North +y
+x East
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?
20 miles South
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.
Fig. 3.4 in your book
R
Basic vector operationsScalar multiplication
3 -3
Multiplying a vector A and a scalar (i.e. number) k makes a vector, denoted by kA.
Intermission
North +y
+x East
50 miles East
30 miles North
Dx
Dy
What is D (the magnitude of )?
A. 58 milesB. 80 milesC. 20 milesD. 0 milesE. 58 m/s
Q20 Think about the Pythagorean Theorem
Vector arithmetic: components
Vector arithmetic: components
A, Ay, and Ax here are the MAGNITUDES of the vectors drawn
(they don’t have hats and are not bold).
AAy
Ax
Vector arithmetic: components
The magnitude of a vector component is its final number
minus initial number!
AAy
Ax
Vector arithmetic: components
|Ax| = Ax = xf - xi
AAy
Ax
|Ay| = Ay
= yf - yi
xfxi
yf
yi
What is the magnitude of the x component of ?
North +y
+x East
Dx
DyA. 60.6 milesB. 35.0 milesC. 40.4 milesD. 0 milesE. 31 degrees
Q21 Think about SOH CAH TOA!
70 miles
30o
Vector arithmetic: components
D
D
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 mathDon’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.
TO START:Draw your vector right triangle.
What are the sides? Compare your triangle with your neighbor.
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?
30 m/s
20°
5.4 m
CAREFUL! You can’t do vector arithmetic combining displacement
(5.4m) with speed (30m/s)!+y
+x