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"due" any direction = toward that direction
Pic 2. Compass rose, showing a vector pointing due east.
Pic 3. Way for describing vector directions.
The vector's direction is written as [E 20° N] - read as "point east and turn 20° to north."
Its complementary angle (2 angles adding to 90°) - 70° is another way to write its direction: [N 70° E] - read as "point north and then turn 70° to east."
Both directions are the same and the word is interchangeable.
Pic 2's compass rose has been used for centuries to describe direction, with applications on land, sea, and air.
Recall that drawn vectors have 2 ends: its tip and tail. Pic 3's vector pointing east in Figure 2 is rotated by 20° to north. A common convention is to represent vectors pointing between the main direction (north, south, east, and west) to show its direction.
convention: way (something is done)
Pic 3 shows how the way can be used to this vector.
Another way used is by using a Cartesian grid, having north and east for the positive y-axis and the positive x-axis respectively.
Vectors added in 2D won't always point due (toward) north, south, east, or west. Like a resultant vector often points at an angle relative to these directions.
resultant vector: a vector made by adding 2 or more given vectors
In a scale like 1 cm : 100 m, think of the ratio as "diagram measurement to real-world measurement."
So a diagram measurement of 5.4 cm = 5.4 x 1 cm represents a real measumrent of 5..4 x 100 m = 540 m.
Problem 1
Problem 1
Joe walks 50 m due east then turns a corner and walks 75 m due north. What's his total displacement?
To add the vectors by scale diagram, a reasonable scale for the diagram must be determined, like 1 cm : 10 m.
The problem can be done by 4 step:
Pic. 5 Vector ∆d, drawn to scale
Drawing the 1st vector (Pic 5): A Cartesian coordinate system is first drawn. Its origin = where x and y-axis cross. The 1st vector is drawn with its tail starting from the origin at 5.0 m (converted by using 1 cm : 10 m ratio from 50 m, 50 ÷ 10) due (to) east.
2. Draw 2nd vector (Pic. 6): The 2nd displacement vector added is 7.5 cm (converted with 1 cm : 10 m, from 75, 75 ÷ 10). 2nd vector's tail is joined to the 1st vector's tip. Vectors are always joined tip to tail.
3. Draw the resultant vector: Pic. 7 shows the resultant vector drawn from 1st vector's tail to the 2nd vector's tip, which are always drawn from the starting point in the diagram (origin) to the ending point and also shows the Angle Theta/Θ formed by it with the 1st vector. The problem is solved by measuring the Angle Theta and the resultant vector and convert it to the true distance with the scale of the diagram.
4. Determine the resultant vectors' magnitude and direction: Either with the Pythagorean Theorem or use a ruler to measure the resultant diagram, it's 9.0 cm, which represents a displacement of 9 x 10 m = 90 m. Angle Theta is 56°. So answer is: 90 m [E 56° N ]
Pic. 7
Pic. 6
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