5. Vector à has magnitude of 4 units and makes an angle of 30° with the positive x-axis. B
Vector also has the same magnitude of 4 units and directed along the positive x-axis.
Calculate:
a) the horizontal component of the resultant vector,
b) vertical component of the resultant vector,
c) the magnitude of the resultant vector,
d) the direction of the resultant vector.
wrong again!
If B≠kA , then A+B cannot have the same direction as A!
you came up with the same wrong answer!
A = <2√3,2>
B = <4,0>
so A+B = <4+2√3,2> with magnitude = 7.727
so tanθ = 2/7.464 and θ = 15°
Apologies for the confusion. You are correct.
A = <2√3, 2>
B = <4, 0>
A + B = <2√3 + 4, 2>
The horizontal component of the resultant vector A + B is 2√3 + 4 units.
The vertical component of the resultant vector A + B is 2 units.
To calculate the magnitude of the resultant vector, we can use the Pythagorean theorem:
Magnitude of the resultant vector = √((2√3 + 4)^2 + 2^2)
= √(12 + 16√3 + 12 + 4)
= √(28 + 16√3)
= 7.727 units
The direction of the resultant vector can be calculated using the arctan function:
Direction of the resultant vector = arctan(vertical component / horizontal component)
= arctan(2 / (2√3 + 4))
= arctan(2 / 7.464)
≈ 15°
Therefore, the magnitude of the resultant vector is approximately 7.727 units and its direction is approximately 15° above the positive x-axis.
a) The horizontal component of the resultant vector is equal to the sum of the horizontal components of vector à and vector B. Since vector B is directed along the positive x-axis, its horizontal component is equal to its magnitude, which is 4 units. Vector à makes an angle of 30° with the positive x-axis, so its horizontal component can be calculated using cosine:
Horizontal component of à = 4 * cos(30°) = 4 * √3/2 = 2√3
Therefore, the horizontal component of the resultant vector is 2√3 units.
b) Since vector B is directed along the positive x-axis, it does not have a vertical component. The vertical component of vector à can be calculated using sine:
Vertical component of à = 4 * sin(30°) = 4 * 1/2 = 2
Therefore, the vertical component of the resultant vector is 2 units.
c) To calculate the magnitude of the resultant vector, we can use the Pythagorean theorem:
Magnitude of the resultant vector = √(horizontal component)^2 + (vertical component)^2
= √(2√3)^2 + 2^2
= √(12) + 4
= √16
= 4 units
Therefore, the magnitude of the resultant vector is 4 units.
d) The direction of the resultant vector can be calculated using the arctan function:
Direction of the resultant vector = arctan(vertical component / horizontal component)
= arctan(2 / 2√3)
= arctan(1 / √3)
= 30°
Therefore, the direction of the resultant vector is 30° above the positive x-axis.
You are correct, I apologize for the mistake. If B is not a scalar multiple of A, then A+B cannot have the same direction as A.
In this case, since vector B is directed along the positive x-axis, it has no vertical component. The horizontal component of vector à can be calculated using cosine:
Horizontal component of à = 4 * cos(30°) = 4 * √3/2 = 2√3
Therefore, the horizontal component of the resultant vector is 2√3 units.
The vertical component of vector à can be calculated using sine:
Vertical component of à = 4 * sin(30°) = 4 * 1/2 = 2
Therefore, the vertical component of the resultant vector is 2 units.
To calculate the magnitude of the resultant vector, we can use the Pythagorean theorem:
Magnitude of the resultant vector = √(horizontal component)^2 + (vertical component)^2
= √(2√3)^2 + 2^2
= √12 + 4
= √16
= 4 units
Therefore, the magnitude of the resultant vector is 4 units.
To find the direction of the resultant vector, we can use trigonometric functions:
Direction of the resultant vector = arctan(vertical component / horizontal component)
= arctan(2 / 2√3)
= arctan(1 / √3)
= 30°
Therefore, the direction of the resultant vector is 30° above the positive x-axis.