Let A and B be the two vectors of magnitude 10 unit . If they are inclined to X-axis at angle 30 degree and 60 degree respectively. Find the resultant.
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To find the resultant of two vectors, we can use vector addition. The resultant vector is the vector sum of the two given vectors.
Given:
Magnitude of vector A = 10 units
Magnitude of vector B = 10 units
Angle between vector A and X-axis = 30 degrees
Angle between vector B and X-axis = 60 degrees
We will first resolve both vectors into their horizontal and vertical components.
For vector A:
Horizontal component of A (Ax) = magnitude of A * cos(angle with X-axis)
Ax = 10 * cos(30) = 10 * sqrt(3)/2 = 5 * sqrt(3) units
Vertical component of A (Ay) = magnitude of A * sin(angle with X-axis)
Ay = 10 * sin(30) = 10 * 1/2 = 5 units
For vector B:
Horizontal component of B (Bx) = magnitude of B * cos(angle with X-axis)
Bx = 10 * cos(60) = 10 * 1/2 = 5 units
Vertical component of B (By) = magnitude of B * sin(angle with X-axis)
By = 10 * sin(60) = 10 * √3/2 = 5 * √3 units
Now, add the horizontal components and vertical components separately to find the resultant components:
Resultant horizontal component (Rx) = Ax + Bx = 5 * sqrt(3) + 5 = (5 * sqrt(3) + 5) units
Resultant vertical component (Ry) = Ay + By = 5 + 5 * √3 = (5 + 5 * √3) units
Finally, find the magnitude of the resultant vector (R) using the Pythagorean theorem:
Magnitude of resultant (R) = sqrt((Rx)^2 + (Ry)^2)
R = sqrt((5 * sqrt(3) + 5)^2 + (5 + 5 * √3)^2) units
Simplifying this equation will give you the exact value of the magnitude of the resultant vector.
Therefore, the magnitude and direction of the resultant vector can be found using the above calculations.
x-axis components add up to
10(cos30 + cos60) = 13.66
y axis components add up to
10(sin 30 + sin60) = 13.66
resultant magnitude = 19.32
The law of cosines could also have
been used.