Three-dimensional geometry: distance, volume, surface area and angles：練習問題

A circle has a radius of \(10\) cm. Find the length of an arc that subtends an angle of \(\frac{2\pi}{5}\) radians at the center.

Given two vectors \( \mathbf{u} \) and \( \mathbf{v} \) such that \( |\mathbf{u}| = 3 \) and \( |\mathbf{v}| = 5 \). If the angle between \( \mathbf{u} \) and \( \mathbf{v} \) is \( 60^{\circ} \), calculate the magnitude of the vector \( 2\mathbf{u} - \mathbf{v} \).

In triangle \(ABC\), the side lengths \(a\), \(b\), and \(c\) are three consecutive integers such that \(a < b < c\). Given that the largest angle, \(C\), is twice the size of the smallest angle, \(A\), find the value of \(a\).

Find the distance between the point A\((1, -2, 3)\) and the point B\((4, 2, 3)\).

In triangle ABC, the side lengths are \(a = 4\), \(b = 13\), and \(c = 15\). Calculate the area of the triangle.

Consider a right-angled triangle where the hypotenuse has a length of 10 cm and one of the acute angles is \(30^{\circ}\). Calculate the length of the side opposite to this angle.

A ship sails 10 km on a bearing of 045° from port A to port B, and then sails 15 km on a bearing of 100° from port B to port C. Calculate the distance from port A to port C.

Given the vectors \(\mathbf{u} = \mathbf{i} + 2\mathbf{j} - \mathbf{k}\) and \(\mathbf{v} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}\), find a unit vector that is perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\).

A circle has a radius of \(6\) cm. A sector of this circle has a central angle of \(\frac{\pi}{3}\) radians.

a) Find the length of the arc of this sector. Give your answer in terms of \(\pi\).

b) Calculate the area of the sector. Give your answer in terms of \(\pi\).

c) Determine the perimeter of the sector. Give your answer in the form \(a + b\pi\), where \(a, b \in \mathbb{Z}\).

In triangle \(ABC\), the side lengths are given as \(AB = 7\), \(BC = 5\), and \(AC = 8\).
a) Calculate the exact magnitude of the angle \(\angle ABC\).
b) Find the exact area of triangle \(ABC\).
c) A point \(D\) is placed on the side \(AC\) such that \(BD\) is the internal angle bisector of \(\angle ABC\). Calculate the exact length of the segment \(BD\).