On fixing the three coordinate axes as unit vectors I,j and k the displacement vector can be written as,
s = 12i + 9j + 8k
So it's magnitude is,
|s| = √(12² + 9² + 8²)
= √289
= 17 cm

**Total Displacement of the Particle**

To find the total displacement of the particle, we need to consider the three given displacements: 12 cm towards the East, 9 cm towards the North, and 8 cm vertically upward. Let's break down the problem step by step:

**Step 1: Displacement towards East**

The particle moves 12 cm towards the East. Since this displacement is only in the horizontal direction, we can represent it as:

Displacement towards East = 12 cm

**Step 2: Displacement towards North**

The particle moves 9 cm towards the North. Since this displacement is only in the vertical direction, we can represent it as:

Displacement towards North = 9 cm

**Step 3: Vertical Displacement**

The particle moves 8 cm vertically upward. This displacement is in the vertical direction, perpendicular to the horizontal displacements. We can represent it as:

Vertical Displacement = 8 cm

Since the vertical displacement is perpendicular to the horizontal displacements, we can use the Pythagorean theorem to find the resultant of the horizontal displacements:

Resultant of horizontal displacements = √((Displacement towards East)^2 + (Displacement towards North)^2)

Plugging in the values:

Resultant of horizontal displacements = √((12 cm)^2 + (9 cm)^2)
= √(144 cm^2 + 81 cm^2)
= √(225 cm^2)
= 15 cm

**Step 4: Total Displacement**

Now, we have the resultant of the horizontal displacements (15 cm) and the vertical displacement (8 cm). To find the total displacement, we can again use the Pythagorean theorem:

Total Displacement = √((Resultant of horizontal displacements)^2 + (Vertical Displacement)^2)

Plugging in the values:

Total Displacement = √((15 cm)^2 + (8 cm)^2)
= √(225 cm^2 + 64 cm^2)
= √(289 cm^2)
= 17 cm

Therefore, the total displacement of the particle is 17 cm.

In summary, to find the total displacement, we first calculated the resultant of the horizontal displacements using the Pythagorean theorem. Then, we used the resultant of the horizontal displacements and the vertical displacement to find the total displacement using the Pythagorean theorem again.