Find the magnitude and direction of this vector using Sine and Cosine Law: 7 m/s on a bearing of 30° and 2 m/s from a bearing of 343.
I assume you want to add the vectors and find the resultant?
I don't mind helping with answers, but I hate to provide the questions as well...
Did you draw a diagram? The angle between the vectors is 47°, so if your vectors are u and v, then if w = u+v,
|w|^2 = 7^2+3^2-2*3*7*cos(47°) = 29.36
so, |w| = 5.42
To find the angle θ between u and w,
sinθ/3 = sin47/5.42
θ = 24°
So w = 5.42@54°
To find the magnitude and direction of the vector, we can use the Sine and Cosine Laws.
First, let's define the given information:
- 7 m/s on a bearing of 30°: This means that the vector has a magnitude of 7 m/s and is directed at an angle of 30° from the positive x-axis.
- 2 m/s from a bearing of 343°: This means that the vector has a magnitude of 2 m/s and is directed at an angle of 343° from the positive x-axis.
To find the sum of these two vectors, we need to resolve them into their x and y components.
For the vector with a magnitude of 7 m/s and a bearing of 30°:
The x-component (Vx) can be calculated using the formula: Vx = V * cos(θ)
So, Vx = 7 * cos(30°)
The y-component (Vy) can be calculated using the formula: Vy = V * sin(θ)
So, Vy = 7 * sin(30°)
Similarly, for the vector with a magnitude of 2 m/s and a bearing of 343°:
The x-component (V'x) can be calculated using the formula: V'x = V' * cos(θ)
So, V'x = 2 * cos(343°)
The y-component (V'y) can be calculated using the formula: V'y = V' * sin(θ)
So, V'y = 2 * sin(343°)
Now that we have the x and y components, we can find the resultant vector by summing the x-components and the y-components separately.
The x-component of the resultant vector (Vr,x) is given by:
Vr,x = Vx + V'x
And the y-component of the resultant vector (Vr,y) is given by:
Vr,y = Vy + V'y
To find the magnitude (Vr) and direction (θr) of the resultant vector, we can use the Pythagorean theorem and inverse tangent function.
The magnitude (Vr) can be calculated using the formula: Vr = sqrt(Vr,x^2 + Vr,y^2)
And the direction (θr) can be calculated using the formula: θr = atan(Vr,y / Vr,x)
Now, substitute the values we calculated into the formulas to find the magnitude and direction of the resultant vector.