Find by drawing and by calculation the resultant of two vectors 3 and 4 units inclined to each other at
A 30 degree
B 90 degree
C 120 degree
so, can you not draw the vectors? Do you have a protractor?
If not, you can use your knowledge of the values of the trig functions of special angles.
B is easy - just the diagonal of the rectangle, right?
I can't
To find the resultant of two vectors, you can use both graphical and mathematical methods. I will explain how to do it for each case you mentioned: A, B, and C.
A) Inclined at 30 degrees:
1. Draw a vector with a length of 3 units starting from an initial point.
2. From the endpoint of the first vector, draw another vector with a length of 4 units at an angle of 30 degrees. Make sure both vectors share the same initial point.
3. Complete the triangle by drawing the third side, which connects the initial point of the first vector with the endpoint of the second vector.
4. The resultant vector is the side of the triangle opposite to the initial point of the first vector. Measure its length and direction.
B) Inclined at 90 degrees:
1. Draw a vector with a length of 3 units starting from an initial point.
2. From the endpoint of the first vector, draw another vector with a length of 4 units at a right angle (90 degrees). Again, make sure both vectors share the same initial point.
3. Combine the vectors by placing the second vector's endpoint at the tip of the first vector.
4. The resultant vector is the straight line formed from the initial point of the first vector to the endpoint of the second vector. Measure its length and direction.
C) Inclined at 120 degrees:
1. Draw a vector with a length of 3 units starting from an initial point.
2. From the endpoint of the first vector, draw another vector with a length of 4 units at an angle of 120 degrees. Ensure both vectors share the same initial point.
3. Combine the vectors by placing the second vector's endpoint at the tip of the first vector.
4. The resultant vector is the line segment that connects the initial point of the first vector with the endpoint of the second vector. Measure its length and direction.
Now, let's calculate the resultants using the mathematical method (vector addition):
A) Inclined at 30 degrees:
1. Resolve the vectors into their horizontal and vertical components.
- Vector A: length = 3 units; angle = 30 degrees
- Vector B: length = 4 units; angle = 180 degrees (opposite direction)
- Vector A horizontal component = A * cos(angle) = 3 * cos(30 degrees)
- Vector A vertical component = A * sin(angle) = 3 * sin(30 degrees)
- Vector B horizontal component = B * cos(angle) = 4 * cos(180 degrees)
- Vector B vertical component = B * sin(angle) = 4 * sin(180 degrees)
2. Add the horizontal and vertical components separately.
- Resultant horizontal component = Vector A horizontal component + Vector B horizontal component
- Resultant vertical component = Vector A vertical component + Vector B vertical component
3. Calculate the magnitude (length) of the resultant vector.
- Resultant magnitude = sqrt(Resultant horizontal component^2 + Resultant vertical component^2)
4. Calculate the direction of the resultant vector.
- Resultant direction = atan(Resultant vertical component / Resultant horizontal component)
Repeat this procedure for cases B) and C) to calculate the magnitudes and directions of the respective resultant vectors.