I.4. Graphical representation of a vector.

To graph a vector we must draw:

• A point of application, called the origin of the vector and denoted by 0;
• A line that gives the direction of the vector;
• The segment of the vector that we measure with the ruler using a standard (the value of the vector is divided by the standard chosen to find out the length of the segment);
• An arrow at the end of the segment that will indicate the direction of the vector.

Let's recap the directions of space, which are infinity. Of these, two are main: vertical and horizontal, the rest up to an infinity are different oblique directions with horizontal and vertical, respectively.

Horizontal direction is the direction given by the surface of a liquid in a vessel at rest. It has two ways: to the right and to the left.

Vertical direction is the direction given by the direction of a lead wire, left free. It has two ways: up and down.

🔦 Remark

The vertical direction is perpendicular to the horizontal direction

The oblique direction is drawn with the help of the protractor. Place the line of the protractor (the one joining 0° by 180°) in the main direction (vertical or horizontal), with its middle at point 0 of the vector and from that direction the angle of the oblique direction with the main one is measured.

East-west direction (E-V) is the direction given by the sun's setting on the celestial vault from east (E) to west (V). It is represented horizontally and has two directions: to E (right) and to V (left).

North-south direction (N-S) is the direction given by the orientation of a magnet left free (suspended by a wire, a needle or water). It is represented vertically and has two directions: N (up) and S (down).

👀 Experiment: Determining the North-South direction with a floating compass

Needed materials:
A bowl of water, cellophane, scissors, needle, fridge magnet.

Experiment description:

• Place the needle on the refrigerator magnet about an hour before performing the experiment and leave it in contact with it to magnetize.
• Fill a bowl with water.
• From a cellophane sheet cut with scissors a sheet 3 cm long and 1 cm wide.
• Place the cellophane sheet on the surface of the water.
• Place the magnetized needle on the cellophane sheet and make sure that the floating needle does not touch the edge of the vessel with water so as not to prevent it from moving freely.
• What do you notice?

  As soon as you place the magnetized needle on the cellophane sheet, it starts to move and after a while it stabilizes in a certain direction.

Experiment conclusion:
The magnetized needle was oriented in the geographic North-South direction of the Earth.

🔓 Solved Problems

1. Graph the following vectors:

1.a. A traveler travels 10 km on a straight road to the east.

Solution:

d = 10 km, horizontal direction, to the right.

Standard: 1 cm : 2 km

To put the point of application we need to see the vector way. This is to the right, we will put 0 to the left of the page.

To find the length of the vector segment we divide the value (modulus) of the vector to the value of the standard: 10 : 2 = 5cm.

We draw a 5 cm segment horizontally and at the top we put the arrow that shows its way. Next to the arrow we note the vector.

1.b. A body is thrown at a speed of 120 km / h, vertically, downwards.

Solution:

So v = 120 km / h, vertical direction, downward way.

Standard: 1 cm : 40 km/h

To put the point of application we need to see the vector way. This being down, we will put 0 at the top of the page.

To find the length of the vector segment we divide the value (modulus) of the vector to the value of the standard: 120 : 40 = 3 cm.

Desenăm un segment de 3 cm pe verticală și în vârf îi punem săgeata care arată sensul lui. Lângă săgeată notăm vectorul.

We draw a 3 cm segment vertically and at the top we put the arrow that shows its way. Next to the arrow we note the vector.

1.c. A force of 1800 N acts on a body, in a direction that makes an angle of 50 ° with the vertical, upwards.

Solution:

So F = 1800 N, direction with an angle of 50 ° with the vertical, upward way.

Standard: 1 cm : 300 N

To put the point of application we need to see the vector way. This being up, we will put 0 at the bottom of the page.

To find the length of the vector segment we divide the value (modulus) of the vector to the value of the standard: 1800 : 300 = 6 cm.

Point the main direction, put the line of the protractor on it, with its middle in 0 and measure the angle of 50° with the vertical. In this oblique direction we draw a 6 cm segment and at the top we put the arrow that shows its way. Next to the arrow we note the vector.

1.d. A body moves accelerated with an acceleration of 42 m/s², in a direction that makes an angle of 30° with the horizontal, upwards.

Solution:

So, a = 42 m/s², in a direction that makes an angle of 30° with the horizontal, upwards.

Standard: 1cm : 6 m/s²

To put the point of application we need to see the vector way. This being up, we will put 0 at the bottom of the page.

To find the length of the vector segment we divide the value (modulus) of the vector to the value of the standard: 42 : 6 = 7 cm.

Dotted the main direction, put the line of the protractor on it, with its middle in 0 and measure the angle of 30° with the horizontal. In this oblique direction we draw a 7 cm segment and at the top we put the arrow that shows its way. Next to the arrow we note the vector.

2. Which of the following physical parameters are scalar or vector, respectively?

• Area (S) - scalar
• Time (t) - scalar
• Acceleration (a) - vector
• Friction force (Ff) - vector
• Temperature (T) - scalar
• Weight force (G) - vector
• Movement (d) - vector