Vector OperationsChapter 3 section 2
A + B = ?
A
B
Vector Dimensions- When diagramming the motion of an
object, with vectors, the direction and magnitude is described in x- and y- coordinates simultaneously.- This allows vectors to be used for 1-d
and 2-d motion.
How can I get to the red dot starting from the origin and can only travel in a
straight line?
x
y
x
y
There are 3 main different ways that I
can travel to get from the origin to the
red dot by only traveling in a straight
lines.
Solving For The Resultant of 2 Perpendicular Vectors
When two vectors are perpendicular to each other it forms a right triangle, when the resultant is formed.
Right triangles have special properties that can be used to solve specific parts of the triangle.Such as the length of sides and angles.
Magnitude of a VectorTo determine the magnitude of two
vectors, the Pythagorean Theorem can be usedAs long as the vectors are perpendicular to
each other.
Pythagorean Theoremc²=a²+b²
(length of hypotenuse)²=(length of leg)²+(length of other leg)²
Applied Pythagorean Theorem
c2=a2+b2 R²=Δy²+Δx² (Mathematics) (Physics)
a
b
c
Δx
ΔyR
Direction of a VectorTo determine the direction of the
vector, use the tangent function.Tangent Function
Tanθ=opp/adj
opp
adjθ
Applied Tangent Function
a=opp
b=adj
c
Δx
ΔyR
θθ
(Mathematics)
(Physics)
Δx
ΔyR
θ
Δx
ΔyR
θ
=
Recall Vector Properties
Example Problem A soldier travels due east for 350
meters then turns due north and travels for another 100 meters. What is the soldiers total displacement?
Example Picture
Example Work
Example AnswerR=364 m @ 15.95°
Vector ComponentsEvery vector can be broken down
into its x and y components regardless of its magnitude or direction.
Vectors Pointing Along a Single Axis
When a vector points along a single axis, the second component of motion is equal to zero.
Vectors That Are Not Vertical or Horizontal
Ask yourself these questions.How much of the vector projects onto
the x-axis?How much of the vector projects onto
the y-axis?
Components of a Vector
x
y
θ
A
A x
A x
Resolving Vectors into Components
Components of a vector – The projection of a vector along the axis of a coordinate system.x-component is parallel to the x-axisy-component is parallel to the y-axisThese components can either be
positive or negative magnitudes.Any vector can be completely
described by a set of perpendicular components.
Vector Component EquationsSolving for the x-component of a
vector.
Solving for the y-component of a vector.
Example ProblemBreak the following vector into its x-
and y- components.A = 6.0 m/s @ 39°
Example Problem WorkA = 6.0 m/s @ 39°
Example Problem AnswerAx = 4.66 m/sAy = 3.78 m/s
Example Problem:A plane takes off from the ground at
an angle of 15 degrees from the horizontal with a velocity of 150mi/hr. What is the horizontal and vertical velocity of the plane?
Example Picture
Example Work
Example AnswerHorizontal velocity = 144.89 miles per
hourVx=144.89mi/hr
Vertical velocity = 38.82 miles per hourVy=38.82mi/hr
Adding Non-Perpendicular Vectors
When vectors are not perpendicular, the tangent function and Pythagorean Theorem can’t be used to find the resultant.Pythagorean Theorem and Tangent only
work for two vectors that are at 90 degrees (right angles)
Non-Perpendicular Vectors To determine the magnitude and direction
of the resultant of two or more non-perpendicular vectors:Break each of the vectors into it’s x- and y-
components. It is best to setup a table to nicely
organize your components for each vector.
Component Tablex-component y-component
Vector A - (A)
Vector B - (B)
Vector C - (C)
Add more rows if needed
Resultant - (R)
Non-Perpendicular Vectors Once each vector is broken into its x- and
y- components :The components along each axis can be added
together to find the resultant vector’s components.Rx = Ax + Bx + Cx + …Ry = Ay + By + Cy + …
Only then can the Pythagorean Theorem and Tangent function can be used to find the Resultant’s magnitude and direction.
Example ProblemDuring a rodeo, a clown runs 8.0m
north, turns 35 degrees east of north, and runs 3.5m. Then after waiting for the bull to come near, the clown turns due east and runs 5.0m to exit the arena. What is the clown’s total displacement?
Practice Problem PictureStep #1: Draw a picture of the
problem
Practice problem WorkStep #2: Break each vector into its x-
and y- components.x-component y-component
Vector A - (A)
Vector B - (B)
Vector C - (C)
Resultant - (R)
Step #3: Find the resultant’s components by adding the components along the x- and y-axis.
x-component y-component
Vector A - (A)
Vector B - (B)
Vector C - (C)
Resultant - (R)+
Step #4: Find the magnitude of the vector by using the Pythagorean
theorem.R2 = Δx2 + Δy2
Step #5: Find the direction of the vector by using the tangent function.
Tan θ = Δy/Δx
Step #5: Complete the final answer for the resultant with its magnitude and
direction.
Practice Problem AnswerResultant displacement = 12.92m @
57.21º