In vector analysis, understanding how to calculate the resultant vector from multiple unit vectors is essential. Let's explore the process of vector addition using three unit vectors, denoted as a, b, and c. Each vector can be expressed in unit vector form, which allows for straightforward addition by combining their respective components.
For the first example, we define the vectors as follows: a = 3i - 3j, b = 1i - 4j, and c = -2i + 5j. To find the resultant vector d, we add the components of each vector:
d = a + b + c = (3 + 1 - 2)i + (-3 - 4 + 5)j = 2i - 2j.
Next, we calculate the magnitude of vector d using the Pythagorean theorem:
|d| = √((2)² + (-2)²) = √(4 + 4) = √8 ≈ 2.83.
To determine the direction, we use the arctangent function:
θd = tan-1(y/x) = tan-1(-2/2) = tan-1(-1) = -45°.
However, since we are interested in the angle's absolute value, we can express it as 45° in the fourth quadrant.
In the second example, we explore a different combination of vectors. Here, we define a new vector e by reversing the direction of vectors a and b and adding vector c. This gives us:
e = -a - b + c = -3i + 3j - (1i - 4j) + (-2i + 5j) = -6i + 12j.
To find the magnitude of vector e, we again apply the Pythagorean theorem:
|e| = √((-6)² + (12)²) = √(36 + 144) = √180 ≈ 13.42.
For the direction of vector e, we calculate:
θe = tan-1(12/-6) = tan-1(-2) ≈ 63.4° (in the second quadrant).
In summary, the process of vector addition involves aligning the vectors, summing their components, and then calculating the resultant vector's magnitude and direction using fundamental trigonometric principles. This method is crucial for solving problems in physics and engineering that involve forces, velocities, and other vector quantities.
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