Vector A,with a magnitude of 23 units, points in the positive x direction. Adding vector B to vector A
yields a resultant vector that points in the negative x direction with a magnitude of 8 units. What are the magnitude and direction of vector
B?
If you add a vector 'B' to vector 'A' such that the resultant is in the direction negative to 'A', then vector 'B' is also in the direction negative to 'A', and has a larger magnitude than A.
When vectors are in the same direction, you can just add them normally to obtain the magnitude of the resultant.
A + (-B) = -8
(The negative signs mean B and the resultant are in the negative direction of A)
=> 23 + (-B) = -8
=> -B = -31
=> B = 31
vector A = (23,0)
vector C = (x,0)
vector B = (-8,0)
v(A) + v(C) = v(B)
(23,0) + (x,0) = (-8,0)
(x,0) = (-31,0)
resultant has magnitude 31 in the negative x direction.
base on my calculation for the direction angle of vector, tan inverse(23/-8)is equal -70degree but the correct answer id 180 degree.Can you explain why 180 degree Arora?
tanθ2 = (Bsinθ/(A + Bcosθ))
In this case,
θ = 180
So, tanθ = 0
And hence, you get the required direction using the formula for angle calculation.
Note that in this case, θ2 is equal to tan-1(0), which is actually zero, not 180.
But because the larger vector is B,
the formula is θ2 = (Asinθ/(B + Acosθ)), which gives you an angle of 0 with B, which means an angle of 180 with A.
A + B = -8.
23 + B = -8,
B = -31 Units = 31 Units, West.
To find the magnitude and direction of vector B, we can break it down into its components along the x-axis.
Since vector A points in the positive x direction and the resultant vector points in the negative x direction, we know that the x-component of vector B must be negative. Let's call this component Bx.
Given that the magnitude of vector A is 23 units and the magnitude of the resultant vector is 8 units, we can use the Pythagorean theorem to determine the magnitude of vector B.
The Pythagorean theorem states that the magnitude of a vector can be found by taking the square root of the sum of the squares of its components. In this case, we have:
Magnitude of vector B = √(Bx^2 + By^2) -> Equation 1
We also know that the x-component of vector B is negative and the resultant vector points in the negative x direction. Therefore, the x-component of vector B is equal in magnitude to the resultant vector, which is 8 units. Thus, Bx = -8.
Substituting Bx = -8 into Equation 1, we have:
Magnitude of vector B = √((-8)^2 + By^2) -> Equation 2
Simplifying Equation 2, we get:
Magnitude of vector B = √(64 + By^2)
Since we are given that the magnitude of vector A is 23 units, we can use this information to solve for By. We know that adding vector B to vector A yields a resultant vector with a magnitude of 8 units. This means that:
Magnitude of vector A + Magnitude of vector B = Magnitude of resultant vector
23 + Magnitude of vector B = 8
Magnitude of vector B = 8 - 23
Magnitude of vector B = -15
Now we have the magnitude of vector B, which is -15.
To find the direction of vector B, we can use the inverse tangent function.
Direction of vector B = tan^(-1)(By / Bx)
Substituting By = Magnitude of vector B = -15 and Bx = -8, we have:
Direction of vector B = tan^(-1)(-15 / -8)
Using a calculator or reference table, we find that the inverse tangent of -15/-8 is approximately 61.9 degrees.
Therefore, the magnitude of vector B is 15 units in the negative x direction, and the direction of vector B is 61.9 degrees in the negative x direction.