find the resultant of vectors A of magnitude 10units in the direction N40E and B of magnitude 8units in the direction S30E
The resultant has a north component of
Y = 10 cos 40 - 8 cos 30 = 0.732
and an east component of
X = 10 sin 40 + 8 sin 30 = 10.43
The resultant's magnitude is
sqrt[(X^2 + Y^2] = 10.46
To find the resultant of two vectors, we need to add them together using vector addition. To do this, we first need to break down each vector into its components using trigonometry.
Let's start with the vector A:
Magnitude of A = 10 units
Direction of A = N40E
To break down the vector A, we need to determine its north-south and east-west components. Using trigonometry, we can find these components.
North component of A = Magnitude of A × sin(angle)
North component of A = 10 × sin(40°)
East component of A = Magnitude of A × cos(angle)
East component of A = 10 × cos(40°)
Calculating these values:
North component of A = 10 × sin(40°) ≈ 6.427 units
East component of A = 10 × cos(40°) ≈ 7.653 units
Now let's move to vector B:
Magnitude of B = 8 units
Direction of B = S30E
Again, we need to find the north-south and east-west components of vector B using trigonometry.
South component of B = Magnitude of B × sin(angle)
South component of B = 8 × sin(30°)
East component of B = Magnitude of B × cos(angle)
East component of B = 8 × cos(30°)
Calculating these values:
South component of B = 8 × sin(30°) ≈ 4 units
East component of B = 8 × cos(30°) ≈ 6.928 units
Now that we have the components of both vectors, we can add them together to find the resultant.
Resultant north component = North component of A - South component of B
Resultant north component = 6.427 - 4 ≈ 2.427 units
Resultant east component = East component of A + East component of B
Resultant east component = 7.653 + 6.928 ≈ 14.581 units
To find the magnitude and direction of the resultant vector, we can use the Pythagorean theorem and trigonometry.
Magnitude of resultant = sqrt((Resultant north component)^2 + (Resultant east component)^2)
Magnitude of resultant = sqrt((2.427)^2 + (14.581)^2) ≈ 14.851 units
Direction of resultant = atan(Resultant north component / Resultant east component)
Direction of resultant = atan(2.427 / 14.581)
Calculating this value:
Direction of resultant ≈ 9.47 degrees (measured clockwise from true north)
Therefore, the resultant of vectors A and B is approximately 14.851 units in the direction N9.47E.