vector A is 5.5cm and points 60°above the positive x-axis. Vector B is 6.5cm and points along thr negative x-axis. Calculate the magnitude and direction of the resultant of these two vectors.
Vx = 5.5 cos 60 - 6.5
Vy = 5.5 sin 60
|V| = sqrt (Vx^2 + Vy^2)
tan theta = Vy/Vx
note in quadrant 3 (draw it)
I mean quadrant 2, negative x, positive y
To calculate the magnitude and direction of the resultant of these two vectors, we first need to find the x and y components of each vector.
Let's start with vector A:
The magnitude of vector A is given as 5.5 cm, and it makes an angle of 60° above the positive x-axis. To find the x component of vector A, we can use trigonometry:
x component of A = magnitude of A * cos(theta)
x component of A = 5.5 cm * cos(60°) = 5.5 cm * 0.5 = 2.75 cm
To find the y component of vector A, we can use trigonometry as well:
y component of A = magnitude of A * sin(theta)
y component of A = 5.5 cm * sin(60°) = 5.5 cm * √(3/2) = 5.5 cm * 0.866 = 4.76 cm
Now let's move on to vector B:
The magnitude of vector B is given as 6.5 cm, and it points along the negative x-axis. Therefore, the x component of vector B will be negative, and the y component will be zero.
x component of B = -6.5 cm
y component of B = 0 cm
Next, we can find the resultant vector by adding the x and y components separately:
x component of resultant = x component of A + x component of B = 2.75 cm + (-6.5 cm) = -3.75 cm
y component of resultant = y component of A + y component of B = 4.76 cm + 0 cm = 4.76 cm
Now, we can calculate the magnitude of the resultant vector using the Pythagorean theorem:
magnitude of resultant = sqrt((x component of resultant)^2 + (y component of resultant)^2)
magnitude of resultant = sqrt((-3.75 cm)^2 + (4.76 cm)^2)
magnitude of resultant = sqrt(14.06 cm^2 + 22.66 cm^2)
magnitude of resultant = sqrt(36.72 cm^2)
magnitude of resultant = 6.06 cm (rounded to two decimal places)
Finally, we can find the direction of the resultant vector using trigonometry:
angle = arctan(y component of resultant / x component of resultant)
angle = arctan(4.76 cm / -3.75 cm)
angle = arctan(-1.2693)
angle = -51.56° (rounded to two decimal places)
Therefore, the magnitude of the resultant vector is 6.06 cm, and it points at an angle of -51.56° (measured counterclockwise from the positive x-axis).