IF A = 2x+3y and B = -x+y, then the angle between the resultant A+B and the positive directio of x-axis is
Ax=2, Bx= - 1 => (A+B)x=2-1=1
Ay=3, By= 1 => (A+B)y=3+1=4
tanα=(A+B)y/(A+B)x =4/1=4,
α =arctan4=76º
To find the angle between the resultant vector A+B and the positive direction of the x-axis, we first need to find the resultant vector A+B.
Given:
A = 2x + 3y
B = -x + y
To find A+B, we add the corresponding components of A and B:
(Ax + Bx) + (Ay + By)
= (2x - x) + (3y + y)
= x + 4y
So, the resultant vector A+B is x + 4y.
Now, to find the angle between the resultant vector A+B and the positive direction of the x-axis, we can use the dot product.
The dot product of two vectors A and B is given by:
A · B = |A| |B| cos(θ)
Where |A| and |B| are the magnitudes of vectors A and B, respectively, and θ is the angle between them.
In this case, we want to find the angle between the resultant vector A+B (x + 4y) and the positive direction of the x-axis.
The positive direction of the x-axis can be represented by the vector i, which has components (1, 0).
So, we can find the dot product of the resultant vector (x + 4y) and the vector i:
(x + 4y) · i = |x + 4y| |i| cos(θ)
Since |i| = 1, we can simplify the equation to:
(x + 4y) · i = |x + 4y| cos(θ)
The dot product of (x + 4y) and i can be found by multiplying their corresponding components:
(x + 4y) · i = x(1) + 4y(0)
= x
So, the equation becomes:
x = |x + 4y| cos(θ)
To solve for θ, we need to find the magnitude of the resultant vector |x + 4y| and substitute it into the equation.
The magnitude of a vector A = (Ax, Ay) is given by:
|A| = sqrt(Ax^2 + Ay^2)
In this case, the magnitude of the resultant vector |x + 4y| is:
|x + 4y| = sqrt((x + 4y)^2)
Substituting this into the equation:
x = sqrt((x + 4y)^2) cos(θ)
Now, we can solve for θ by rearranging the equation:
cos(θ) = x / sqrt((x + 4y)^2)
To find the angle θ, take the inverse cos (cos^−1) of both sides of the equation:
θ = cos^−1(x / sqrt((x + 4y)^2))
This is the angle between the resultant vector A+B and the positive direction of the x-axis.