You walk 25 m south and 40 m east. Find the magnitude and direction of the resultant displacement both graphically and algebraically.
Magnitude is 47.16
What is the degree?
the angle S of E is arcTangent(25/40)
X = 40 m.
Y = -25 m.
Tan A = Y/X = (-25)/40,
A = -32o = 32o S. of E. = 328o CCW.
To find the direction of the resultant displacement, we need to calculate the angle it forms with the reference direction (usually the positive x-axis).
First, let's find the angle using trigonometry:
The angle can be found using the tangent function:
tan(θ) = opposite/adjacent
where opposite = 25 m (south), and adjacent = 40 m (east).
θ = arctan(opposite/adjacent)
θ = arctan(25/40)
Using a calculator, we find:
θ ≈ 32.5 degrees
Therefore, the direction of the resultant displacement is approximately 32.5 degrees east of south.
To find the direction of the resultant displacement, we can use trigonometry.
Algebraically:
1. Start by plotting the two displacement vectors on a coordinate plane.
- The displacement of 25 m south can be represented as (-25, 0) on the coordinate plane.
- The displacement of 40 m east can be represented as (40, 0) on the coordinate plane.
2. Add the two vectors to find the resultant displacement.
- Add the x-components: -25 + 40 = 15
- Add the y-components: 0 + 0 = 0
3. The resultant displacement is (15, 0) on the coordinate plane.
4. Use the Pythagorean theorem to find the magnitude of the resultant displacement.
- Magnitude = √(15^2 + 0^2) = √225 = 15
5. To find the direction, we can use the inverse tangent (arctan) function.
- Degree = arctan(0/15) = arctan(0) = 0°
Therefore, the degree of the resultant displacement is 0°.
Note: Since the y-component is zero, the resultant displacement is along the x-axis, which means it is directly eastward.