a man walks 10km towards west and then 15 km towards south. Find the resultant vectors, magnitude and direction.
X = hor. = -10km.
Y = ver. = -15km.
tanAr = y/x = -15 / -10 = 1.50,
Ar = 56.3 deg. = reference angle,
A = Ar + 180 = 56.3 + 180 = 236.3 deg.
R = X / cosA = -10 / cos236.3 = 18km @
236.3 deg., Q3.
To find the resultant vector, magnitude, and direction, we can use the concept of vector addition.
Step 1: Assign directions to the given distances.
- 10 km towards the west can be represented as -10 km towards the east (because west is opposite to east).
- 15 km towards the south remains unchanged.
Step 2: Represent the distances as vectors.
- The vector towards the west is denoted as (-10 km, 0 km).
- The vector towards the south is denoted as (0 km, -15 km).
Step 3: Add the vectors.
- To find the resultant vector, simply add the corresponding components of the vectors.
- (-10 km, 0 km) + (0 km, -15 km) = (-10 km, -15 km).
Step 4: Calculate the magnitude of the resultant vector.
- The magnitude of a vector can be found using the Pythagorean theorem: magnitude = √(x^2 + y^2), where x and y are the components of the vector.
- magnitude = √((-10 km)^2 + (-15 km)^2)
- magnitude = √(100 km^2 + 225 km^2)
- magnitude = √(325 km^2)
- magnitude ≈ 18.03 km (rounded to two decimal places).
Step 5: Calculate the direction of the resultant vector.
- The direction of a vector can be found using trigonometry.
- The angle (θ) of the vector can be calculated using the tangent function: θ = arctan(y/x), where y and x are the components of the vector.
- θ = arctan((-15 km)/(-10 km))
- θ = arctan(1.5)
- θ ≈ 56.31 degrees (rounded to two decimal places).
Therefore, the resultant vector is approximately (-10 km, -15 km) with a magnitude of approximately 18.03 km and a direction of approximately 56.31 degrees.