Intro to Vectors
• Vectors indicate direction; scalars do not.
• Vectors are represented by symbols
• Vectors can be added graphically
Properties of Vectors
• Vectors can be moved parallel to themselves in a diagram
• Vectors can be added in any order
• To subtract a vector, add its opposite
• ****Multiplying or dividing vectors by scalars results in a
Head to Tail Method
• To add/subtract two vectors you MUST ALWAYS have the
vectors in a head to tail formation.
• The red vector, is the sum of the two other vectors, we call
this vector the RESULTANT vector, R.
• Show visually how these two vectors, A & B are added.
The red line is the
resultant of the two
vectors, R = A+B
Mathematical Vector Addition
• We can calculate MAGNITUDE of the RESULTANT of two
vectors mathematically by using the pythagorean
• We can calculate the DIRECTION of the RESULTANT of
two vectors mathematically by using the tangent function.
• Vector A = 3 m
• Vector B = 4 m
• What is A+B?
An archaelogist climbs the Great Pyramid in Giza, Egypt. If
the pyramids height is 136 m and its width is 2.30 x 102 m,
what is the magnitude and the direction of the
archaelogist’s displacement while climbing from the bottom
of the pyramid to the top?
While following the directions on a treasure map, a pirate
walks 45 m north, then turns and walks 7.5 m east. What
single straight-line displacement could the pirate have
taken to reach the treasure?
Resolving Vectors into Components
• If you are given the RESULTANT vector of two vectors
then we can find the components of each of these vectors
by using the sine and cosine functions for right triangles
• Sinθ = opp/hyp
• Cosθ= adj/hyp
• Find the component velocities of a helicopter traveling 95
km/hr at an angle of 35o to the ground.
• How fast must a truck travel to stay beneath an airplane
that is moving 105 km/hr at an angle of 25o with the
Multiply Vectors by a Scalar
• When you multiply any vector by a scalar the result is
always a vector
• Ex: If vector A = 15.2m at 65o, What is the value of 3A?
• What about 4A?
Adding Vectors that are not Perpendicular
• To add vectors that are not perpendicular, you must find
the components of each of those vectors and add them
and then find the Resultant of their added components.
• A hiker walks 25.5 km from her base camp at 35o south of
east. On the second day, she walks 41 km in a direction
65o north of east, at which point she discovers a forest
rangers tower. Determine the magnitude and direction of
her resultant displacement between the base camp and
the ranger’s tower.
• A football player runs directly down the field for 35 m
before turning to the right at an angle of 25o from his
original direction and running an additional 15 m before
getting tackled. What is the magnitude and direction of
the runner’s total displacement?
• Until now, we have only been dealing with motion in one
dimension, now we will start working in 2 dimensions
• The use of components (x-direction and y-direction)
avoids vector multiplication
• Components SIMPLIFY projectile motion
• We neglect air resistance AND the rotation of the Earth so
therefore ---- Projectiles follow parabolic paths
• IN GENERAL, Projectile Motion is free fall with an initial
How to Solve Projectiles
• First step: Create a chart that looks as follows… FOR
EVERY QUESTION YOU SOLVE
Variable X Y
Vf (final velocity)
X or Y (displacement)
THINGS TO KNOW
• ALWAYS START WITH THE Y-DIRECTION … BECAUSE
Y NOT? Lol (but really)
• TIME IN THE X = TIME IN THE Y, EVERYTIME.
In the y-direction…
• Write your variables as follows: voy, vfy, ay, Δy & t
• MOTION IN THE Y-DIRECTION IS IN FREE FALL, meaning our
objects are moving at the acceleration due to gravity…….
• ay = - 9.8 m/s2, ALWAYS!!!!!!!!!!!!!!!!!!!!
In the x-direction …
• Write your variables as follows: vox, vfx, ax, Δx & t
• MOTION IN THE X-DIRECTION IS CONSTANT, meaning our
velocity is constant the entire time so that means …..
• ax = 0 m/s2, ALWAYS!!!!!!!!!!!!!!!!!!!!
One more thing …
•X-direction and y-direction can be
treated and solved independently.
Projectiles launched horizontally
• When projectiles are launched horizontally, the following
is true of your variables….
Variable X Y
Vf (final velocity) 0 m/s
a (acceleration) 0 m/s2 -9.8 m/s2
X or Y (displacement)
It is launched only in
the horizontal, so
your initial velocity in
the y is 0.
Example (pg. 101 Sample 3D)
• The Royal Gorge Bridge in Colorado rises 321 m above
the Arkansas River. Suppose you kick a little rock
horizontally off the bridge. The rock hits the water such
that the magnitude of its horizontal displacement is 45 m.
Find the speed at which the rock was kicked.
You try! (pg. 102 #1)
• An autographed baseball rolls off of a 0.70 m high desk
and strikes the floor 0.25 m away from the base of the
desk. How fast was it rolling?
Projectile launched at an angle
• When projectiles are launched at an angle you now have
components of your initial velocity
GROUND or SURFACE
Example 2 (pg. 104 #3)
• A baseball is thrown at an angle of 25o relative to the
ground at a speed of 23.0 m/s. If the ball was caught 42.0
m from the thrower, how long was it in the air? How high
was the tallest spot in the ball’s path?
You try! (pg 104 #2)
• A golfer can hit a golf ball a horizontal distance of over
300 m on a good drive. What maximum height will a
301.5 m drive reach if it is launched at an angle of 25o to
the ground? (Hint: At the top of its flight, the ball’s vertical
velocity component will be zero)