1. Introduction to Vectors
Introduction to Vectors
2D Kinematics I

2. Vectors & Scalars A scalar quantity has magnitude but no direction
Examples:
speed
volume
temperature
mass
A vector is a quantity that has both magnitude and direction
displacement
velocity
acceleration
force

3. Vectors in One Dimension
With Motion in 1D, our vectors could point in only two possible directions:
positive
Negative
Direction was indicated by the sign (+/-)
Examples: +10 m/s meant “to the right” or “up”
 9.81 m/s2 meant “to the left” or “down”

4. Vectors in Two Dimensions  y x
In 2D motion, objects can move
up-down and left-right
north-south and east-west
Direction must now be specified as an angle
Vectors (drawn as arrows) can represent motion
The length of the arrow is proportional to the magnitude
of the vector

5. Vector Addition
Vectors can slide around in the plane without changing, but...
Changing the magnitude changes the vector
Changing the direction changes the vector
To add two vectors graphically:
Slide them (without changing them) until they are “tip-to-tail”
The resultant vector is from the tail of the first to the tip of the second

6. Adding Perpendicular Vectors
To add perpendicular vectors we can:
do it graphically or
use Pythagorean theorem and the tangent function
Let’s add two perpendicular displacements, x and y, to get the resultant displacement d
x = 1.5 m right
y = 2.0 m down y x x  y
tan  = —— x y
 = tan–1(——) x y d
 = tan–1(———)
1.5 m
-2.0 m
= -53

7. Resolving Vectors into Components
The vectors x and y are the component vectors of the vector d
The x-component vector is always parallel to the x-axis
The y-component vector is always parallel to the y-axis x y d

8. cos  =  =  sin  =  =  x = d cos  y = d sin 
We can resolve d back into its components using the cos  and sin  functions:
adj
hyp x d
cos  =  = 
adj d y x 
opp
hyp y d
sin  =  = 
opp
hyp
x = d cos 
y = d sin 
x = d cos  = (2.5 m) cos (-53°) = 1.5 m
y = d sin  = (2.5 m) sin (-53°) = -2.0 m

9. The components of d can be represented by dx and dy
dx = x = 1.5 m
dy = y = -2.0 m
Notice dy is given a negative sign to indicate that y is pointing down

10. Example of Components
Find the components of the velocity of a helicopter traveling 95 km/h at an angle of 35° to the ground

11. 𝒗 𝒙 =𝟗𝟓 𝐜𝐨𝐬 𝟑𝟓° =𝟕𝟕.𝟖𝒌𝒎/𝒉
𝒗 𝒚 =𝟗𝟓 𝐬𝐢𝐧 𝟑𝟓° =𝟓𝟒.𝟓𝒌𝒎/𝒉

12. Add the following vectors
𝑨 =𝟐.𝟓𝒎/𝒔 𝜽=𝟒𝟓°
𝑩 =𝟓.𝟎𝒎/𝒔 𝜽=𝟐𝟕𝟎°
𝑪 =𝟓.𝟎𝒎/𝒔 𝜽=𝟑𝟑𝟎°
Hint: Vectors have both a magnitude and direction. θ is the direction.

13. 𝑨 𝒙 =𝟐.𝟓 𝐜𝐨𝐬 𝟒𝟓°=𝟏.𝟕𝟕𝒎/𝒔 𝑨 𝒚 =𝟐.𝟓 𝐬𝐢𝐧 𝟒𝟓° =𝟏.𝟕𝟕𝒎/𝒔
𝑨 𝒙 =𝟐.𝟓 𝐜𝐨𝐬 𝟒𝟓°=𝟏.𝟕𝟕𝒎/𝒔 𝑨 𝒚 =𝟐.𝟓 𝐬𝐢𝐧 𝟒𝟓° =𝟏.𝟕𝟕𝒎/𝒔
𝑩 𝒙 =𝟓.𝟎 𝒄𝒐𝒔 𝟐𝟕𝟎°=𝟎𝒎/𝒔 𝑩 𝒚 =𝟓.𝟎 𝒔𝒊𝒏 𝟐𝟕𝟎° =−𝟓.𝟎𝟎𝒎/𝒔
𝑪 𝒙 =𝟓.𝟎 𝒄𝒐𝒔 𝟑𝟑𝟎°=𝟒.𝟑𝟑𝒎/𝒔 𝑪 𝒚 =𝟓.𝟎 𝒔𝒊𝒏 𝟑𝟑𝟎° =−𝟐.𝟓𝟎𝒎/𝒔
𝑹 𝒙 = 𝑨 𝒙 + 𝑩 𝒙 + 𝑪 𝒙 =𝟔.𝟏𝟎𝒎/𝒔 𝑹 𝒚 = 𝑨 𝒚 + 𝑩 𝒚 + 𝑪 𝒚 =−𝟓.𝟕𝟑𝒎/𝒔
𝑹= 𝑹 𝒙 𝟐 + 𝑹 𝒚 𝟐 =𝟖.𝟑𝟕𝒎/𝒔
𝜽= tan −𝟏 𝑹 𝒚 𝑹 𝒙 =𝟑𝟏𝟕° 𝐨𝐫 −𝟒𝟑.𝟐°