Find the resultant of two vectors of 3 units and 6units acting at a point of 0 at an angle of 45° with each other
To find the resultant of two vectors acting at an angle, we can use the parallelogram rule.
Let's call the first vector A and the second vector B.
Given that A has a magnitude of 3 units and B has a magnitude of 6 units, and they act at an angle of 45 degrees with each other, we can represent A and B as:
A = 3 units (at an angle of 0 degrees)
B = 6 units (at an angle of 45 degrees)
To find the resultant, we need to find the sum of A and B.
Step 1: Convert the angles to their corresponding components.
The x-component of A (Ax) = magnitude of A (3) * cos(angle of A (0)) = 3 * cos(0) = 3 * 1 = 3
The y-component of A (Ay) = magnitude of A (3) * sin(angle of A (0)) = 3 * sin(0) = 3 * 0 = 0
The x-component of B (Bx) = magnitude of B (6) * cos(angle of B (45)) = 6 * cos(45) = 6 * (√2/2) = 6 * (√2)/2 = 3√2
The y-component of B (By) = magnitude of B (6) * sin(angle of B (45)) = 6 * sin(45) = 6 * (√2/2) = 6 * (√2)/2 = 3√2
Step 2: Find the sum of the x-components and the sum of the y-components.
Sum of the x-components = Ax + Bx = 3 + 3√2
Sum of the y-components = Ay + By = 0 + 3√2
Step 3: Find the magnitude and direction of the resultant using the Pythagorean theorem and trigonometry.
Magnitude:
Resultant magnitude = √[(Sum of the x-components)^2 + (Sum of the y-components)^2]
= √[(3 + 3√2)^2 + (0 + 3√2)^2]
= √[9 + 18√2 + 18 + 18√2 + 18]
= √[45 + 36√2] units
Direction:
Resultant direction = tan^(-1)(Sum of the y-components / Sum of the x-components)
= tan^(-1)((0 + 3√2) / (3 + 3√2))
= tan^(-1)(√2 / (1 + √2))
= tan^(-1)(√2 * (1 - √2) / (1 - (√2)^2))
= tan^(-1)(√2 * (1 - √2) / (1 - 2))
= tan^(-1)(-√2 * (√2 - 1))
≈ -35.26° (rounded to two decimal places)
Therefore, the resultant of the two vectors with magnitudes 3 units and 6 units, acting at an angle of 45 degrees with each other, is approximately √[45 + 36√2] units at an angle of -35.26°.