VectorsUnit SIX, Lesson 6.3By Margielene D. Judan

LESSON OUTLINE

Vector RepresentationGraphical MethodMathematical MethodPythagorean TheoremComponent Method

Various quantities in nature can be:

• Scalar quantity – magnitude only

• Vector quantity – magnitude + direction

The length of the arrow represents the magnitude while the angle where the arrow is pointed represents the direction.

Thus, a longer vector has a larger magnitude.

A B CWhich among the vectors has the largest magnitude?

Examples of Vector Representation:

• 30 km/hr East

• 20 km/hr West

• 10 km/hr North

30 km

20 km

10 km

Examples of Vector Representation:

• 50 km/hr NE

• 50 km/hr SE

50 km

50 km

Vectors can also be drawn in a Cartesian coordinate system.

West

North

East

South

Ex. A force of 80 Newtons east

= 80 N eastW

N

E

S

Ex. A velocity of 120 km/hr southwest

= 120 km/hr SW

𝜃=45 °

𝜃=45°

W

N

E

S

Ex. A displacement of 100 m 30 north of west

= 100 m 30 N of W

𝜃=30 °W

N

E

S

We use a protractor to measure the angles in degrees, and a ruler to measure the magnitude.

Vector Addition

We can add vectors using different methods. The sum of the vectors (vector sum) is called the resultant vector, denoted by R.

Vector Addition Methods

• Graphical Method or the Tip-to-Tail Method

• Mathematical Method

1. Pythagorean Theorem

2. Component Method

1. Graphical Method or Tip-to-Tail Method

Tip-to-Tail Method

Arrow 1 Arrow 2

Arrow 1

Arrow 2

Resultant

Resultant

Review from Lesson 6.1

It is called tip-to-tail because you connect the arrows from tip to tail

Correct Wrong

tiptail

tail

tiptip

tailtail

tail

tail

tip

tip

tip

We can use many arrows.

2. Mathematical Method

Look at the red line. Can you measure its exact length?

Using the graphical method is easy and convenient. However, it does not predict measurements exactly. You cannot measure 62.5213 on a protractor and 2.617 cm in a ruler exactly. Thus, we use a more exact method called the mathematical method.

Note: Right and Up (+)Left and Down (-)

Note: Right and Up (+)Left and Down (-)

3. Pythagorean Theorem

Note: The Pythagorean Theorem is used for determining the resultant of two vectors that makes a right angle to each other.The formula is given below:

Use the Pythagorean Theorem to determine the resultant vector below.

Practice A Solution:

Practice B Solution:

AssignmentAnswer Laboratory 3.2.

4. Component Method

Most of the times, the vectors given do not form right angles and the Pythagorean Theorem is not applicable.

Pythagorean Theorem applicable

Pythagorean Theorem not applicable

The Component Method is the best method to use in all vector problems which vectors do not form a right angle (Pythagorean Theorem).Using this method, vectors are broken down into its x and y components.

Given the vector (black), find its x and y components using the graphical method.

Answer: x-component (blue), y-component (red)

x-component

y-componentResultant (R

)

After breaking the vector into its x and y components, we could now apply the Pythagorean Theorem to measure the resultant.

x-component

y-componentResultant (R

)

Using the mathematical method, however, we have to apply concepts in trigonometry.

Given: 50 m, 40 N of E. Find its x and y components.

𝜃=40 °

50 m

Given: 50 m, 40 N of E. Find its x and y components.

𝜃=40 °

dx – x component dy – y component

50 m

Given: 50 m, 40 N of E. Find its x and y components.

𝜃=40 °

dx = d cos dy = d sin

50 m

Given: 50 m, 40 N of E. Find its x and y components.

𝜃=40 °

50 m

dx = d cos dx = 50 cos 40dx = 38.30 m

dy = d sin dy = 50 sin 40dy = 32.14 m

Given: 50 m, 40 N of E. Find its x and y components.

𝜃=40 °

32.14 m50 m

38.30 m

Find the Resultant Using Component Method (Steps)

1. Make a graphical model of the vectors.

2. Find the x and y components of each vector.

3. Find the sum of all x-components and all y-components.

4. Use the Pythagorean Theorem to find the magnitude of the resultant.

5. Find the direction using tan =

Example:

Arrow 1 = 3 km 30 N of EArrow 2 = 4 km 60 S of W

1. Make a graphical model of the vectors.

Arrow 1 = 3 km 30 N

Arrow 2 = 4 km 60 S of W

𝜃=30 °𝜃=60 °

Note: Right and Up (+)Left and Down (-)

+y

+x

-y

-x

2. Find the x and y components of each vector.

Arrow 13 km 30 N

Ax = d cos Ax = -3 cos 30Ax = -2.60 km

Ay = d sin Ay = -3 sin 30Ay = -1.5 km

Arrow 24 km 60 S of W

Bx = d cos Bx = 4 cos 60Bx = 2 km

By = d sin By = 4 sin 60By = 3.46 km

Negative because x is to the left

Negative because y is downward

3. Find the sum of all x-components and all y-components.

x-component total Rx = Ax + Bx

= -2.60 km + 2 km

= -0.6 km

y-component total Ry = Ay + By

= -1.5 km + 3.46 km

= 1.96 km

4. Use the Pythagorean Theorem to find the magnitude of the resultant.

(magnitude of resultant)

5. Find the direction using tan =

tan = tan = tan = -3.27To find , remove tan by typing tan-1(-3.27) in the calculator. (teacher will teach you how) = -73.00

Answer: The displacement is2.05 km, -73.00 Removing the negative sign:2.05 km, 73.00 S of E

𝜃=73 °

Whiteboard Work:

Arrow 1 = 20 km 40 S of EArrow 2 = 10 km 60 N of E

Assignment: (short coupon bond)

From Calapan City Port, the ship travels 15 km, 30 N of W and 10 km, 10 E of N before reaching Batangas City Port. Calculate the displacement between the ports. How far did the ship travel? Make an illustration by drawing. (30 pts)Note: Box your final answers.

Sources:

Science Links 7