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Understanding the concept of vector components is crucial when dealing with movement in different directions, much like the journey described from one city to another. A vector is something that has both magnitude and direction.

• The journey is made up of three vectors: north, northwest, and east.
• Each vector can be broken down into its components, which align with the north-south and east-west axes.

For instance, the northwest journey needs to be split into north and west components since it forms a 45-degree angle with the north direction. This splitting is accomplished using trigonometric calculations, such as applying the sine and cosine for a 45-degree angle, which in this case are both equal. By doing this, each segment of the journey is adjusted to fit into our regular compass directions, making it easier to perform further calculations.

The Pythagorean Theorem is a fundamental principle in mathematics that allows us to calculate the magnitude of the displacement when directional journeys form a right triangle. In the context of our car journey, the theorem is utilized after all vectors are resolved into the north-south and east-west axes.

• The theorem states that in a right-angled 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.
• We have a displacement formed by `north` as one side and `west` as the other.

When we know the components as 117 km north and 22 km west, they form the two sides of the right triangle. The displacement magnitude is therefore the hypotenuse, which can be calculated using:\[ \text{Displacement magnitude} = \sqrt{117^2 + 22^2} \]Plugging in the values will give the resultant distance between the starting and ending points, which serves as the substitution for the hypotenuse.

Directional vectors give us a clearer understanding of not only how far the car travels but also in which direction it travels. When dealing with vectors, it’s not enough to simply know the distance; the direction is equally important.

• Vectors in opposite directions, like east-west or north-south, can affect the total distance differently.
• By calculating direction, we can effectively manage the components, such as knowing how much distance is actually gained or lost.

In the exercise, we start by isolating different direction components: 'north', together with 'northwest', and 'east'. Considering these vectors allows us to determine how each part of the journey contributes to the final displacement. Ultimately, the subtle interaction between these components translates into the accurate computation of the car's journey path, allowing a precise summary of its displacement magnitude as seen with the use of vectors.