Final answer:
To find the acute angle that the given line makes with the positive direction of the x-axis, we can use the concept of direction angles.
Using the direction cosines provided, we can express them as ratios of the lengths of the sides of a right triangle and solve for the angle using trigonometric relationships. The resulting acute angle is approximately 35.26 degrees.
Step-by-step explanation:
To find the acute angle that the line with direction cosines 1/√3, 1/√6, n makes with the positive direction of the x-axis, we can use the concept of direction angles.
The direction angle of a vector is the angle it forms with the positive direction of a reference axis. In this case, we are looking for the angle between the line and the positive x-axis.
Since the direction cosines of the line are given as 1/√3, 1/√6, n, we can express these direction cosines as ratios of the lengths of the sides of a right triangle.
Let's call the acute angle between the line and the positive x-axis as theta. So, cos(theta) = 1/√3 and sin(theta) = 1/√6.
Using trigonometric relationships, we can solve for the value of theta. Taking the inverse cosine of 1/√3, we find theta ≈ 35.26 degrees.
Therefore, the acute angle that the line with direction cosines 1/√3, 1/√6, n makes with the positive direction of the x-axis is approximately 35.26 degrees.