PHYS101: Introduction to Mechanics

3a. Add and subtract vectors

• Describe the properties of vectors. How do you identify a vector in a problem? 
• Define the commutative and associative properties of vectors. 
• Use the head-to-tail graphical method to add vectors.
• Use the head-to-tail graphical method to subtract vectors.

A vector is a quantity that has both a magnitude (amount) and direction. Often in texts, vectors are denoted by being bolded, and/or by having a small arrow written above the vector name.

For example, a vector called A can be written as A or as $\overrightarrow{A}$. The magnitude, or amount, of the vector A equals the value of A. We can think of vectors as arrows, with the length being the magnitude of the vector, and the arrow pointing in the direction of the vector.

Review an example of how a vector can be represented as an arrow in Figure A.1.1 Vectors as Arrows.

When adding or subtracting vectors, we can follow many of the rules we learned in math class about non-vector numbers.

Vector addition follows the commutative property, which means the order of addition does not matter. Vector addition also follows the associative property, which means it does not matter which vector is first when vectors are being added.

Review a complete list of vector addition, subtraction, multiplication, and division rules at the beginning of page A-3 of Vector Analysis. Note that the commutative and associative properties are the most useful.

One way to add or subtract vector is to do so graphically. The graphical method for adding and subtracting vectors is called the head-to-tail method.

When adding vectors using this method:

1. Draw the first vector starting from the tail, or starting point of the vector, to its head, or ending point (arrow) of the vector.
2. Begin the second vector by putting its tail at the head of the first vector.
3. Finally, draw a line from the tail of the first vector to the head of the second vector.

The vector that results is the resultant vector, or the solution to the vector addition problem. To determine the magnitude of the resultant vector, measure it with a ruler. To determine the direction of the resultant vector, use a protractor to determine the angle.

Review an example of using the head-to-tail method to add two vectors, and the step-by-step figures below it, in Figure 3.14.

Review a worked example of using the head-to-tail method to add multiple vectors in Example 3.1: Adding Vectors Graphically Using the Head-to-Tail Method: A Woman Takes a Walk.

When subtracting vectors graphically, consider the vector that is being subtracted as negative. That means the direction of the vector being subtracted is flipped so it points in the opposite direction. The head-to-tail process is the same as it is for addition.

Review a worked example of using the head-to-tail method to subtract vectors in Example 3.2: Subtracting Vectors Graphically: A Woman Sailing a Boat.

3b. Determine the components of a vector given its magnitude and direction, and determine the magnitude and direction of a vector given its components

• Explain the advantages of analytical methods for adding and subtracting vectors, as compared to graphical methods.
• Define component vectors.
• Explain how to use right triangles to resolve a vector into its components. What equations do you use to obtain the x and y coordinate component vectors?
• If you are given component vectors, what equations do you use to determine the magnitude and angle of the resultant vector?

The other method for adding and subtracting vectors is by using analytical methods. Analytical methods use trigonometry to solve vector addition and subtraction. While we still use arrows to represent vectors, analytical methods reduce the measurement errors that can happen with graphical (head-to-tail) methods. 

To use analytical methods to solve vector problems, we need to resolve vectors into their component vectors in the x–y coordinate system.

Review an example of a vector that has been resolved into its x and y components in Figure 3.26. Here, the vector A has a magnitude A and an angle 𝛳. The vector can be broken up into its two components: A_x and A_y. We know that A_x + A_y = A. However, we must use our knowledge of trigonometry to determine how the scalar or magnitude part of each vector relates to each other. The magnitudes of the component vectors relate to the resultant vector the following way:

$A_x=A\cos\theta$

$A_y=A\sin\theta$

Review an example of a vector that has been resolved into its component vectors and shows the magnitudes of the component vectors in Figure 3.27.

Sometimes you are given the component vectors and need to determine the magnitude and angle of the resultant vector. To do this, we again use the trigonometry of right triangles:

$A=\sqrt{A_x^2+A_y^2}$

$\theta=\tan^{-1}\left( \frac{A_y}{A_x} \right )$

Review an example of a resultant vector that is calculated from its component vectors in Figure 3.28.

3c. Separately analyze the horizontal and vertical motions in projectile problems

• Define projectile motion, projectile, trajectory. What is the assumption we make when we calculate projectile motion?
• What are the steps for solving a projectile motion problem?
• Solve projectile motion problems.

We define projectile motion as the motion of a thrown object that only feels the acceleration of gravity. The projectile is the object being thrown, and the trajectory is the path the object takes when it is thrown. When performing projectile motion calculations, we assume there is no air resistance, so gravity is the only force acting on the projectile.

Projectile motion problems are vector problems. We need to know the vector of the trajectory, and then resolve the vector into its components. The steps for solving these problems are the following:

• Step 1: Resolve the trajectory vector into its x and y component vectors.
• Step 2 and 3: Calculate the equations of motion for each component separately. Review the equations we use for the x and y components of the trajectory vector in equations 3.34, 3.35, 3.37, 3.38, 3.39, and 3.40 in Projectile Motion.
• Step 4: Recombine the component vectors into a new resultant vector to calculate total displacement and velocity. Review the equations we use to recombine the component vectors in equations 3.41, 3.42, 3.43, and 3.44.

Review worked examples of performing these calculations using the four-step process listed above in Example 3.4: A Fireworks Projectile Explodes High and Away and Example 3.5: Calculating Projectile Motion: Hot Rock Projectile.

Unit 3 Vocabulary

• Analytical methods
• Angle
• Associative property
• Commutative property
• Component vector
• Graphical methods
• Head-to-tail method
• Magnitude
• Projectile
• Projectile motion
• Resultant vector
• Right triangle
• Trajectory
• Vector
• Vector head
• Vector tail