Question

oabcde is a regular hexagon of side 2 units in the xy plane o being the origin and oa taken along the x -axis. a point p is taken on a line parallel to z -axis through the centre of hexagon at a distance of 3 units from o . the vector ap is

Answer 1

The vector AP, from point A to point P in a regular hexagon with a side of 2 units, where P is 3 units above the centre of the hexagon, is calculated to be (-1, sqrt(3), 3).

Step-by-step explanation:

To find the vector AP, we need to consider the coordinates of both A and P. Since OA is along the x-axis and the hexagon is regular with side 2 units, we know the coordinates of A are (2,0,0). The centre of the hexagon is located at the midpoint of its side, which is also its radius when inscribed in a circle.

Thus, we can find the x and y coordinates of the centre by drawing perpendicular bisectors from the vertices to the opposite sides.

Given the hexagon's symmetry and the fact that regular hexagons can be subdivided into equilateral triangles with height equal to the radius of the circumscribed circle, this distance is sqrt(3) times the side length. Therefore, the x and y coordinates of the centre, O', are (1, sqrt(3), 0). Since point P is 3 units above O' in the z-direction, the coordinates of P become (1, sqrt(3), 3).

Using these coordinates for A (2, 0, 0) and P (1, sqrt(3), 3), we can find the vector AP by subtracting the coordinates of A from those of P, resulting in the vector components given by (1 - 2, sqrt(3) - 0, 3 - 0), which simplifies to (-1, sqrt(3), 3).