91Ó°ÊÓ

When dealing with vectors, magnitude and direction are pivotal in understanding their influence and combining them accurately. Vectors can be represented graphically as arrows, where each vector has a length corresponding to its magnitude and an angle corresponding to its direction.
Magnitude refers to the size or length of the vector. For instance, if vector \( \overrightarrow{\mathbf{A}} \) has a magnitude of 63 units, it means the vector's length is 63 units.
Direction is often specified relative to a reference direction, like due north, east, south, or west. In our exercise, \( \overrightarrow{\mathbf{A}} \) points due west, and \( \overrightarrow{\mathbf{B}} \) points due south. These cardinal directions help us set up a coordinate system to manage further calculations like addition and subtraction.

The Pythagorean Theorem is a handy tool for calculating the resultant vector when two vectors are perpendicular to each other, forming a right triangle. The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
For vectors \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \) that are perpendicular, their sum or difference forms the hypotenuse of a right triangle.
The formula is:

• \[ R = \sqrt{A^2 + B^2} \]

Here, \( A \) and \( B \) are the magnitudes of the vectors. In our exercise, both magnitudes are 63 units. Applying the formula gives us the magnitude of the resultant vector, approximately 89.10 units, whether we add or subtract them.

Creating a right triangle with vectors helps in visualizing vector addition and subtraction. When vectors are placed tail to head in the coordinate system, and if they are perpendicular, they naturally form a right triangle.
In this setup, the path from the start of the first vector to the end of the second vector (in this case from \( \overrightarrow{\mathbf{A}} \) to \( \overrightarrow{\mathbf{B}} \)) gives us the resultant vector.
Think of it this way:

• The west-going vector is the horizontal leg of the triangle.
• The south- or north-going vector is the vertical leg.
• The hypotenuse of this triangle is the resultant vector.

Employing right triangle methods, like the Pythagorean Theorem, simplifies finding both the magnitude and angle of the resultant.

Trigonometric functions, like tangent, help determine the direction of the resultant vector by using the ratio of the opposite side to the adjacent side in a right triangle.
In our exercise, to find the direction, we use:

• \[ \theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) \]

When adding vectors \( \overrightarrow{\mathbf{A}} \) (west) and \( \overrightarrow{\mathbf{B}} \) (south), the angle \( \theta \) tells us how much below the west vector the resultant is.
Here, the angle \( \theta = \tan^{-1}(1) \), which is \( 45^\circ \). This particular angle indicates equal influence by both vectors, forming an angle that splits the difference between their directions, producing a vector directed 45 degrees south of west when added.
Similarly, for subtraction, it results in 45 degrees north of west, showing the directional shift when changing vector directions by subtracting.