find the resultant and direction of2 vectors of 3units and4unit acting at a point o at an angle of 45 with each other with cosine rule
Using the cosine rule, we can find the magnitude of the resultant vector:
$r^2 = 3^2 + 4^2 - 2(3)(4)\cos(45^\circ)$
$r^2 = 25 - 12\sqrt{2}$
$r \approx 1.86$
To find the direction of the resultant vector, we can use the sine and cosine functions:
$\theta = \sin^{-1}\left(\frac{3\sin(45^\circ)}{r}\right)$
$\theta \approx 26.57^\circ$
Therefore, the resultant vector has a magnitude of approximately 1.86 units and points in a direction of approximately 26.57 degrees (measured counterclockwise from the positive x-axis).
To find the resultant and direction of two vectors acting at a point, we can use the cosine rule. The cosine rule states that the square of the magnitude of the resultant is equal to the sum of the squares of the magnitudes of the two vectors plus twice the product of the magnitudes of the vectors and the cosine of the angle between them.
Let's calculate the resultant:
1. Given vectors:
- Vector A: 3 units
- Vector B: 4 units
2. Angle between the vectors: 45 degrees
Using the cosine rule, we can calculate the magnitude (R) of the resultant vector:
R^2 = A^2 + B^2 + 2AB * cos(angle)
R^2 = 3^2 + 4^2 + 2 * 3 * 4 * cos(45)
R^2 = 9 + 16 + 24 * cos(45)
R^2 = 25 + 24 * 0.7071
R^2 = 25 + 16.9704
R^2 = 41.9704
Taking the square root of both sides, we find:
R ≈ √41.9704
R ≈ 6.4768 units
So, the magnitude of the resultant vector is approximately 6.4768 units.
Now, let's find the direction of the resultant vector:
To determine the direction, we can use the inverse tangent function (tan^-1) with the y-component and x-component of the resultant vector.
Let's assume the x-component of the resultant vector is A, and the y-component is B. Then:
tan(angle) = B / A
tan(angle) = 4 / 3
angle ≈ tan^-1(4 / 3)
angle ≈ 53.1301 degrees
Therefore, the direction of the resultant vector is approximately 53.1301 degrees.
To summarize:
- Magnitude of the resultant vector: Approximately 6.4768 units
- Direction of the resultant vector: Approximately 53.1301 degrees