Find the resultant of two vectors of a unit and if unit acting art a point art an angle 45° of each other
To find the resultant of two vectors that are acting at an angle of 45° to each other, we can use vector addition.
Let's denote the two vectors as A and B, with magnitude 1 (unit vector). Since they are at an angle of 45° to each other, we can assume they share the same origin point.
To find the resultant vector, we can add the two vectors using their individual components. The components of vector A and B are given as follows:
Vector A: A = (A_x, A_y) = (1*cos(45°), 1*sin(45°)) = (√2/2, √2/2)
Vector B: B = (B_x, B_y) = (1*cos(45° + 90°), 1*sin(45° + 90°)) = (-(√2/2), √2/2)
Now, we can add the components of A and B to obtain the resultant vector. Adding the x-components and y-components separately, we get:
Resultant vector R: R = (R_x, R_y) = (A_x + B_x, A_y + B_y) = (√2/2 - √2/2, √2/2 + √2/2) = (0, √2)
Therefore, the resultant vector R has a magnitude of √2 and acts along the positive y-axis.
Note: The value of √2 is an approximation.
To find the resultant of two vectors of unit magnitude, we can use the parallelogram law or the triangle law of vector addition.
Using the parallelogram law, we can draw two vectors starting from the same point and forming a parallelogram. The diagonal of the parallelogram is the resultant vector.
The magnitude of each vector is 1, and they are acting at an angle of 45 degrees with respect to each other. Since they are both of unit magnitude, we can consider them as the vectors (1, 0) and (0, 1) in the x-y coordinate system.
Let's name the first vector A and the second vector B.
A = (1, 0)
B = (0, 1)
Using the parallelogram law, we can create the parallelogram by copying vector B and placing the tip of it at the tip of vector A. After drawing the parallelogram, the diagonal represents the resultant vector.
To find the resultant vector, we can use the formula:
Resultant = vector A + vector B
Using vector addition, we add the corresponding components of vectors A and B:
Resultant = (1 + 0, 0 + 1)
= (1, 1)
Therefore, the resultant vector is (1, 1).
To find the magnitude of the resultant vector, we can use the Pythagorean theorem:
Magnitude of Resultant = √(x^2 + y^2)
= √(1^2 + 1^2)
= √2
Therefore, the magnitude of the resultant vector is √2.
To find the resultant of two vectors, each with a magnitude of 1 and acting at an angle of 45° from each other, we can use vector addition.
Step 1: Draw a diagram
Draw a coordinate system and represent the two vectors using arrows. The first vector, denoted as vector A, is pointing in the positive x-direction, and the second vector, denoted as vector B, is at an angle of 45° from vector A.
Step 2: Resolve the vectors into their x and y components
Since the magnitudes of both vectors are 1, their x and y components will also have magnitudes of 1.
For vector A:
- x-component (Ax): 1
- y-component (Ay): 0
For vector B:
- x-component (Bx): 1 * cos(45°) = √2 / 2
- y-component (By): 1 * sin(45°) = √2 / 2
Step 3: Add the x and y components separately
To find the resultant vector, add the x-components and the y-components separately.
For the x-component: Ax + Bx = 1 + √2 / 2 = (2 + √2) / 2
For the y-component: Ay + By = 0 + √2 / 2 = √2 / 2
So, the resultant vector is (√2 / 2, (2 + √2) / 2).