An ant crawls 23 m to the north, then turns 30° towards the right and crawls another 23 m. Determine the direction of the ant's displacement vector, expressed as an angle relative to east.

D = 23[90o] + 23[30o].

D = 23i + 19.9+11.5i = 19.9 + 42.9i = 47.3m[65.1o].

Direction = 65.1o N. of E.

Correction: D = 23i + 19.9+11.5i = 19.9 + 34.5i = 39.8m[60.0o].

Direction = 60.0o N . of E.

To determine the direction of the ant's displacement vector, we can use trigonometry.

First, let's consider the initial northward crawl of 23 m. This creates a vertical displacement of 23 m.

Next, the ant turns 30° towards the right and crawls another 23 m. This creates a diagonal displacement.

To find the horizontal displacement, we can use the trigonometric functions sine and cosine. Since the ant turned to the right, we need to use cosine. The horizontal displacement can be calculated as:

Horizontal displacement = 23 m * cos(30°)

Using the value of cos(30°) from a calculator, we find:

Horizontal displacement ≈ 23 m * 0.866 = 19.918 m

So, the horizontal displacement is approximately 19.918 m.

Now we have both the vertical and horizontal displacements. To find the displacement vector's angle relative to east, we can use the inverse tangent function (arctan) to calculate the angle:

Angle = arctan(vertical displacement / horizontal displacement)

Angle = arctan(23 m / 19.918 m)

Using a calculator, we find:

Angle ≈ 50.19°

Therefore, the ant's displacement vector is at an angle of approximately 50.19° relative to east.

To determine the direction of the ant's displacement vector, we can break down the ant's movement into two separate vectors: the northward movement and the rightward movement.

Let's start with the northward movement. The ant crawls 23 m to the north, so we can represent this vector as 23 m directly up (or in the positive y-axis direction).

Next, let's consider the rightward movement. The ant turns 30° towards the right and crawls another 23 m. Since the ant turned towards the right, we need to find the horizontal component of this vector.

To find the horizontal component, we can use trigonometry. The angle between the horizontal component and the 23 m vector is 30°. So, the horizontal component can be calculated as follows:

Horizontal component = 23 m * cos(30°)

Using the cosine function, we find that the horizontal component is approximately 19.92 m.

Now, we have the two components of the ant's displacement vector: 19.92 m horizontally (positive x-axis direction) and 23 m vertically (positive y-axis direction).

To determine the direction of the displacement vector relative to east, we can use the tangent function. The tangent of an angle is equal to the ratio of the vertical component to the horizontal component.

In this case, the angle we want to find is the one between the displacement vector and the positive x-axis (east). So, we can use the formula:

Angle = arctan(vertical component / horizontal component)

Angle = arctan(23 m / 19.92 m)

Using a calculator, we find that the angle is approximately 50.66°.

Therefore, the direction of the ant's displacement vector, expressed as an angle relative to east, is around 50.66°.