The velocity of an aiplane heading east at 400 knots and a wind velocity of 50 knots northeast. The airplane encounters this wind, show the resultant velocity. What is the resulant speed and direction of the airplane? (the direction is the angle the resultant makes with due north measured clockwise from north)

It is easy to do graphically.

Analytically, make each vector with its components.
Vairplane=400E + O N
Vair=50cos45E + 50sin45N

add the vectors.

I don't understnad how you found any of the equations.

The first Vairplane was given.

The second is given, however I used trig to convert it to N, E components.

http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html#vec5

could you explain the last equations we have not yet learned that.

To find the resultant velocity of the airplane, we need to consider the vector sum of the airplane's velocity and the wind velocity.

Step 1: Break down the velocities into their components.
The airplane's velocity is heading east, which means it has no component in the northward direction and a full component in the eastward direction.
The wind velocity is northeast, which means it has components in both the northward and eastward directions.

Given:
Airplane velocity = 400 knots (eastward)
Wind velocity = 50 knots (northeast)

Let's break them down into components:

Airplane velocity:
Eastward component = 400 knots
Northward component = 0 knots

Wind velocity:
Eastward component = 50 knots * cos(45°) (since northeast is 45 degrees angle from east)
Northward component = 50 knots * sin(45°) (since northeast is 45 degrees angle from east)

Eastward component of wind velocity = 50 knots * cos(45°) = 50 knots * √2 / 2 = 25√2 knots
Northward component of wind velocity = 50 knots * sin(45°) = 50 knots * √2 / 2 = 25√2 knots

Step 2: Find the resultant components.
To find the resultant components, we'll add the corresponding components together.

Resultant eastward component = Airplane eastward component + Wind eastward component
= 400 knots + 25√2 knots

Resultant northward component = Airplane northward component + Wind northward component
= 0 knots + 25√2 knots

Step 3: Find the resultant speed.
To find the resultant speed, we'll use the Pythagorean theorem.

Resultant speed = √((resultant eastward component)^2 + (resultant northward component)^2)
= √((400 knots + 25√2 knots)^2 + (25√2 knots)^2)

Step 4: Find the angle the resultant makes with due north clockwise.
To find this angle, we'll use trigonometry.

Angle = arctan((resultant eastward component) / (resultant northward component))

Now, let's calculate the values:

Resultant eastward component = 400 knots + 25√2 knots = 400 knots + 25 * 1.414 knots ≈ 435.12 knots
Resultant northward component = 25√2 knots = 25 * 1.414 knots ≈ 35.36 knots

Resultant speed = √((435.12 knots)^2 + (35.36 knots)^2) ≈ 437.39 knots

Angle = arctan((435.12 knots) / (35.36 knots)) ≈ 86.64°

Therefore, the resultant speed of the airplane is approximately 437.39 knots, and the angle the resultant makes with due north measured clockwise from north is approximately 86.64°.