What is the position vector that represents the velocity of the plane relative to the ground?

Question

A small plane is flying horizontally due east in calm air at 200 ​mi/hr when it is hit by a horizontal crosswind blowing southwest at 40 mi/hr and a 20 mi/hr updraft. Find the resulting speed of the plane and describe with a sketch the approximate direction of the velocity relative to the ground.

Let the unit vectors i​, j​, and k point​ east, north, and​ upward, respectively. Begin by writing vectors describing the velocity of the​ plane, the​ crosswind, and the updraft. What is the position vector that represents the velocity of the plane relative to the​ ground?

Find the magnitude of this velocity vector to determine the speed of the plane with the wind.

Answer 1

This is an exercise in interpreting the statement

"the unit vectors i​, j​, and k point​ east, north, and​ upward, respectively".

"200 mi/h due east" means 200i

"40 mi/h due southwest" means velocity that is at 45° between

the negative direction of i and negative direction of j.

The length of the vector is 40, and its projections on the i and j axes

are sin(45°) and cos(45°), respectively, both of which are 1/√2.

Therefore "40 mi/h due southwest" means -40i/√2 -40j/√2

"20 mi/h updraft" means 20k

The resulting position vector of the velocity is the vector sum of all three:

200i - 40i/√2 - 40j/√2 + 20k = (200 - 40/√2)i - 40j/√2 + 20k

The speed is the Euclidean norm (magnitude) of the vector:

√((100 - 40/√2)² + (-40/√2)² + 20²) = 79.6 mi/h