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