Vectors

VECTORS

A. VECTOR QUANTITIES AND SCALAR QUANTITIES

In studying Physics, you will encounter scalar and vector quantities. Here are examples of these quantities:

Vectors

1. Displacement: A boy runs 200 m northward.

2. Velocity: A car moves 80 km/h, 40° east of north.

3. Force: A force of 100 N acts on a body in the southward direction.

Scalars

1. Mass: A load has a mass of 50 kg.

2. Time: The girl has traveled 45 minutes.

3. Distance: The train has moved a distance of 40 km.

Difference between Vector and Scalar Quantities

• Vector Quantities: These are quantities indicating both the magnitude (a numerical value along with a unit of measure) and the direction.

• Scalar Quantities: These are quantities that only indicate the magnitude, without any directional attribute.

Representation of Vectors

An arrow is the symbol used to denote a vector quantity. A vector has three important parts:

1. Arrowhead: Indicates the direction of the vector.

2. Length of the Arrow: Represents the magnitude of the vector.

3. Tail: Points to the origin of the vector.

VECTORS

B. RESULTANT VECTOR

The sum of two or more vectors is represented by a single vector called the resultant vector. It is defined as the combination of all the given vectors, obtained either graphically or mathematically derived. A resultant vector is said to represent the shortest possible path between the start and end points of the vectors involved.

GRAPHICAL METHOD

This method employs measuring instruments such as a metric ruler and a protractor to determine the resultant vector.

• The ruler measures the magnitude of the given and resultant vector, while the protractor measures both the direction and the angle of the given and resultant vectors.

• Scaling: This entails assigning scales to represent vectors in graphical methods.

Steps in Graphical Method

1. Choose an appropriate scale and frame of reference for the given vectors.

2. Draw the first vector starting from the origin of the reference frame.

3. Draw the second vector starting from the head of the first vector.

4. Continue drawing the remaining vectors starting from the head of the most recent vector.

5. All vectors must be connected in a head-to-tail pattern.

6. Draw a new vector connecting the tail of the first vector to the head of the last vector plotted. This new vector is the resultant vector of the given vectors.

2. COMPONENT METHOD

The component method is a more convenient and accurate method to add vectors. This method involves mathematically or analytically solving the components of every provided vector. It applies to two or more vectors.

Steps in Component Method

1. Draw and plot the vectors graphically without assigning a scale as it will be solved mathematically.

2. Resolve the X and Y components.

3. Get the algebraic sum of the vectors using the equations ext{X} and ext{ΣY}.

4. Compute for the magnitude of the resultant vector using the Pythagorean Theorem:
   R = \sqrt{X^2 + Y^2}

5. Find the angle of direction using the equation:
   heta = an^{-1}\left(\frac{y}{x}\right)

Shortcut method for determining Trigonometric Functions

1. If the angle of a vector is from the x-axis, the X component is computed as:

   • X = ext{cos(θ)} * ext{Magnitude}

   • The Y component is computed as:

   • Y = ext{sin(θ)} * ext{Magnitude}

2. If the angle of a vector is from the Y-axis:

   • The X component is computed as:

   • X = ext{sin(θ)} * ext{Magnitude}

   • The Y component can be computed as:

   • Y = ext{cos(θ)} * ext{Magnitude}

VECTORS

Example Problem

A disoriented physics professor drives 5.18 km east, then 6.20 km north, and finally 2.05 km west. The problem involves finding the magnitude and direction of the resultant displacement using the components method.

Vectors and their Components