Vector A has magnitude 3.6 units; vector B has magnitude 5.9 units. The angle between A and B is 45degrees. What is the magnitude of A + B?
To find the magnitude of A + B, we can use the concept of vector addition. Vector addition involves combining the magnitudes and directions of two vectors.
First, let's determine the components of vectors A and B. Since the angle between A and B is 45 degrees, we can use basic trigonometry to find the horizontal and vertical components of each vector.
The horizontal component of vector A can be found using the formula:
Horizontal Component = Magnitude × cos(angle)
Substituting the given values, we have:
Horizontal Component of A = 3.6 × cos(45°) = 3.6 × √2/2 = 3.6 × 0.707 = 2.546
Similarly, the vertical component of vector A can be found using the formula:
Vertical Component = Magnitude × sin(angle)
Substituting the values:
Vertical Component of A = 3.6 × sin(45°) = 3.6 × √2/2 = 3.6 × 0.707 = 2.546
Now, let's calculate the horizontal and vertical components of vector B in the same way.
Horizontal Component of B = 5.9 × cos(45°) = 5.9 × √2/2 = 5.9 × 0.707 = 4.169
Vertical Component of B = 5.9 × sin(45°) = 5.9 × √2/2 = 5.9 × 0.707 = 4.169
To find the horizontal and vertical components of A + B, we can add the respective horizontal and vertical components of vectors A and B.
Horizontal Component of A + B = Horizontal Component of A + Horizontal Component of B = 2.546 + 4.169 = 6.715
Vertical Component of A + B = Vertical Component of A + Vertical Component of B = 2.546 + 4.169 = 6.715
Now, we can find the magnitude of A + B using the Pythagorean theorem:
Magnitude of A + B = √(Horizontal Component of A + B)^2 + (Vertical Component of A + B)^2
Substituting the given values:
Magnitude of A + B = √(6.715)^2 + (6.715)^2 = √(45.077225) + (45.077225) = √90.15445 ≈ 9.495 units
Therefore, the magnitude of A + B is approximately 9.495 units.