Projectile motion:
Projectile motion refers to the motion of an object that is projected into the air and is subject only to the force of gravity and air resistance (if present). The object follows a curved path known as a parabola.

Velocity of projection:
The velocity of projection of a projectile is the initial velocity at which the object is launched. It has both horizontal and vertical components.

Given that the velocity of projection of the projectile is (6i 8j) m/s, where i and j are the unit vectors in the x and y directions respectively.

Horizontal range:
The horizontal range of a projectile is the total horizontal distance covered by the projectile before it hits the ground. It can be calculated using the formula:

Range = (2 * v * sinθ) / g

Where v is the magnitude of the velocity of projection, θ is the angle of projection, and g is the acceleration due to gravity.

However, in this case, we are not given the angle of projection. So, we need to find it using the given velocity components.

Calculating the angle of projection:
The angle of projection can be calculated using the formula:

tanθ = (vertical component of velocity) / (horizontal component of velocity)

tanθ = (8j) / (6i)

tanθ = (8/6) * (j/i)

tanθ = (4/3) * (2/1) * (j/i)

tanθ = (8/3) * (j/i)

Therefore, the angle of projection is tan⁻¹(8/3).

Calculating the horizontal range:
Now that we have the angle of projection, we can substitute the values into the formula to calculate the horizontal range.

Range = (2 * v * sinθ) / g

Range = (2 * √(6² + 8²) * sin(tan⁻¹(8/3))) / g

Range = (2 * √(36 + 64) * (8/3) / √(64/9)) / g

Range = (2 * √(100) * (8/3) / (8/3)) / g

Range = (2 * 10 * (8/3)) / g

Range = (160/3) / g

Range = (160/3) / 9.8

Range ≈ 17.35 meters

Therefore, the horizontal range of the projectile is approximately 17.35 meters.