To solve this problem, we need to consider the vector components of the aircraft's landing speed and the crosswind. The goal is to determine the course the aircraft should steer to counteract the crosswind and land on the runway aligned at 263 degrees.

1. Aircraft's Landing Speed Vector:

   • The aircraft's intended landing speed is 72 m/s in the direction of 263 degrees.

2. Crosswind Vector:

   • The crosswind is 15 m/s blowing due west. In navigation terms, "due west" corresponds to 270 degrees.

3. Vector Addition:

   • To land safely in the direction of 263 degrees, the aircraft needs to adjust its heading to counter the effect of the crosswind. This requires vector addition of the aircraft's velocity vector and the crosswind vector.

4. Calculating Necessary Heading:

   • Let's set up a coordinate system where:

     • North is 0 degrees,
     • East is 90 degrees,
     • South is 180 degrees,
     • West is 270 degrees.

   • The crosswind vector can be broken into components:

     • ( V_{\text{crosswind, x}} = 15 , \text{m/s} \cdot \cos(270^\circ) = 0 )
     • ( V_{\text{crosswind, y}} = 15 , \text{m/s} \cdot \sin(270^\circ) = -15 , \text{m/s} )

   • The aircraft's desired trajectory along the runway is at 263 degrees. To counteract the crosswind, we need to find the correct heading angle where the resultant velocity vector points at 263 degrees.

5. Using Vector Addition:

   • Let ( \theta ) be the heading angle the aircraft needs to steer.

   • The velocity components of the aircraft relative to the air are:

     • ( V_{\text{aircraft, x}} = 72 , \text{m/s} \cdot \cos(\theta) )
     • ( V_{\text{aircraft, y}} = 72 , \text{m/s} \cdot \sin(\theta) )

   • The resultant velocity components should satisfy the runway alignment:

     • ( V_{\text{resultant, x}} = V_{\text{aircraft, x}} + V_{\text{crosswind, x}} )
     • ( V_{\text{resultant, y}} = V_{\text{aircraft, y}} + V_{\text{crosswind, y}} )

   • We want the resultant vector to be in the direction of 263 degrees:

     • ( \tan(263^\circ) = \frac{V_{\text{resultant, y}}}{V_{\text{resultant, x}}} )

6. Solving for (\theta):

   • Simplifying the equations and solving for (\theta) will give the required heading. This step involves trigonometric calculations that can be done using a calculator or computational tools.

   • After solving, you should find that the aircraft should steer slightly into the wind to counteract it, resulting in a course slightly less than 263 degrees.

In conclusion, the exact heading angle (\theta) should be calculated using the trigonometric relationships described, ensuring that the resultant vector aligns with the runway heading of 263 degrees.