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When we talk about finding the speed of an object that is moving in two different directions, we're really talking about finding the magnitude of its velocity vector. Imagine a blimp flying both north and east at the same time. The velocity components in the north and east directions can be visualized as the two sides of a right triangle. The blimp's actual speed, which is the hypotenuse of this triangle, is the magnitude of its velocity.
To calculate this magnitude, use the Pythagorean theorem, which tells us that the square of the length of the hypotenuse (here, the speed) equals the sum of the squares of the other two sides (the velocity components). Thus, the formula is:

• \( v = \sqrt{v_{north}^2 + v_{east}^2} \)

For a blimp with velocities of 15 km/h north and 15 km/h east, the speed is

• \( v = \sqrt{15^2 + 15^2} = \sqrt{450} \approx 21.21 \text{ km/h} \)

This gives us the blimp's overall speed, calculated by combining its movements in both directions.

Figuring out which direction an object is moving involves finding the angle its path makes with a known line, like the north direction. For a blimp moving north and east, we want to determine the angle of its path from north. This direction is determined using the inverse tangent (arctan or \( \tan^{-1} \)) of the ratio of the eastward velocity to the northward velocity.
Here's the formula used:

• \( \theta = \tan^{-1} \left( \frac{v_{east}}{v_{north}} \right) \)

When both velocities are equal, as in our example where both are 15 km/h, the angle is:

• \( \theta = \tan^{-1}(1) = 45^{\circ} \)

Therefore, the direction of motion of the blimp is at an angle of 45 degrees east of north, forming a perfect diagonal between the north and east directions.

The Pythagorean theorem is a fundamental part of mathematics used to calculate distances and understand relations in right-angled triangles. In vector addition, it becomes crucial for finding the magnitude of a vector when given its components. The theorem states that for 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.
In our context, this theorem helps in finding the speed of a blimp traveling at velocity components in perpendicular directions like north and east. Specifically, if \( v_{north} \) and \( v_{east} \) are the legs of the triangle, the Pythagorean theorem gives:

• \( c^2 = a^2 + b^2 \)

Translating this to vectors, the speed \( v \) is:

• \( v = \sqrt{v_{north}^2 + v_{east}^2} \)

This theorem is a powerful tool that allows us to combine the perpendicular components of velocity to find an object's resultant speed in a straightforward manner.