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Vectors

Allow me to introduce Despicable Me’s Vector

Vector Despicable Me

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Vectors have X and Y components

Mathematically, the components act like shadows of the force vector on the coordinate axes.

Here we see a force vector on the (x, y) plane. The force vector is white, the x-axis is red, the y-axis is green, the origin is white.

We force vectors with their tails at the origin.

The light is shining directly into the (x, y) plane. We see no shadows from this view.

Below is the same scene from another viewpoint. The light is now shining directly from above (shining straight down, parallel to the y-axis.)

Note the shadow of the vector on the x-axis: it represents the x-component of the force vector.

Next we have the same situation except the direction of the light has changed.

The light now is shining from the right, parallel to the x-axis.

A shadow of the force vector can be seen on the y-axis.

This shadow, mathematically, is the y-component of the force vector.

We are back to a flat surface diagram below; it shows how these components can be drawn.

The black vector is the two dimensional force vector, labeled F.

The red vector is the x-component of the force vector, labeled Fx.

It is pronounced ‘F sub x’. Since ‘x’ is a subscript, it really looks like this:

The subscript’s position is often implied, as here:

The green vector is the y-component of the force vector, labeled Fy, pronounced ‘F sub y’.

The components of the force vector can also be arranged this way, forming a right triangle:

forceComponents3

The sign of the components

The x-component of the force vector can be positive or negative.

  • If it points to the right, it is positive.

  • If it points to the left, it is negative.

The y-component of the force vector can be positive or negative.

  • If it points up, it is positive.

  • If it points down, it is negative.

When right triangle trigonometry is used, use your vector diagram to decide which way the components are pointing. Then assign the correct sign to your values, as a last step in your solution.

The right triangle trigonometry as presented here will always yield positive results. It is for finding the lengths of the legs of a right triangle, as one might do in geometry.

Using sailboats

One important application of this principle is in the recreational sport of sail boating. A full lesson is presented here

Lesson 1 Vector Components

Lesson Resolution of Forces

also see Points of Sail

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The horizontal (sideways) component of motion is independent of the vertical component of the motion. This isn’t intuitive to everyone, so sometimes we’ll see a motion that doesn’t match our gut instincts. For instance, what is happening here?

Let’s see another example of how vertical and horizontal motion are independent.

When we get to the unit on projectile motion we can study this in more depth.

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How to draw a parabola

How to create a parabola GIF

The Physics of Doing an Ollie on a Skateboard, or, the Science of Why I Can’t Skate

Firing a projectile on a flat surface

PhET Projectile Motion lab!

Projectile Motion app: Galileo and Einstein

Firing a projectile on a curved surface…like the Earth!

Newton’s Mountain Canon: Galileo and Einstein

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Vectors – Motion and Forces in Two Dimensions

  1. Lesson 1 – Vectors: Fundamentals and Operations

    1. Vectors and Direction

    2. Vector Addition

    3. Resultants

    4. Vector Components

    5. Vector Resolution

    6. Component Addition

    7. Relative Velocity and Riverboat Problems

    8. Independence of Perpendicular Components of Motion

  2. Lesson 2 – Projectile Motion

    1. What is a Projectile?

    2. Characteristics of a Projectile’s Trajectory

    3. Horizontal and Vertical Components of Velocity

    4. Horizontal and Vertical Displacemen

    5. Initial Velocity Components

    6. Horizontally Launched Projectiles – Problem-Solving

    7. Non-Horizontally Launched Projectiles – Problem-Solving

  3. Lesson 3 – Forces in Two Dimensions

    1. Addition of Forces

    2. Resolution of Forces

    3. Equilibrium and Statics

    4. Net Force Problems Revisited

    5. Inclined Planes

    6. Double Trouble in 2 Dimensions

Vectors and Projectiles

Vector Addition

Drag a vector onto the canvas. Drag the arrowhead to change its direction. Repeat up to two more times and guess the direction of the resultant. Improve your skill at adding vectors using the head-to-tail method.

Name that Vector

The Name That Vector Interactive is a skill building tool that presents users with 12 vector addition challenges. Twenty-five vectors are displayed on a grid; each challenge involves adding three of the vectors together to determine the resultant.

Vector Guessing Game

The Vector Guessing Game will challenge learner’s understanding of adding vectors.Two random vectors are displayed and learners must decide on the size and direction of the resultant.

Vector Addition: Does Order Matter?

The Vector Addition: Does Order Matter? Investigate with this app

Projectile Simulator

The variable-rich environment of the Projectile Simulator Interactive allows a learner to explore a variety of questions associated with the trajectory of a projectile. Learners can modify the launch height, the launch angle, and the launch speed and observe the effect upon the trajectory.

The Monkey and the Zookeeper

Learning Standards

Massachusetts Science Curriculum Framework 2016

1. Motion and Forces: Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.1 Compare and contrast vector quantities (e.g., displacement, velocity, acceleration force, linear momentum) and scalar quantities (e.g., distance, speed, energy, mass, work).

Common Core Math
Expressing Geometric Properties with Equations
CCSS.MATH.CONTENT.HSG.GPE.A
Translate between the geometric description and the equation for a conic section

Common Core Standards for Mathematics – Grades 9-12
Standards for Mathematical Practice:

  • Reason abstractly and quantitatively
  • Model with mathematics
  • Look for and express regularity in repeated reasoning
Number and Quantity – Vector and Matrix Quantities
  • N-VM.1     Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments and use appropriate symbols for vectors and their magnitudes.
  • N-VM.2     Find the components of a vector.

 

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