\((1 / \mathrm{c}),(1 / \mathrm{c}),(1 / \mathrm{c})\) is direction cosine of the line then the value of \(\mathrm{c}=\) (a) \(\pm(\overline{1 / 3)}\) (b) \(\pm 3\) (c) \(\pm(1 / \sqrt{3})\) (d) \(\pm \sqrt{3}\)

Vector algebra is a critical component of mathematics, especially when dealing with the geometry of space. It provides us with tools to represent various quantities, such as forces, velocity, and displacement, which have both magnitude and direction. Within this framework, the concept of direction cosines emerges as a way to quantify the orientation of a vector in relation to the coordinate axes.

Direction cosines are, in essence, cosines of angles that a line or vector makes with the positive directions of the x, y, and z axes. There are three direction cosines for a line in three-dimensional space, usually denoted as \(l\text{, } m\text{, } n\), which correspond to the cosines of the angles between the line and the x, y, and z axes, respectively. A critical property of these direction cosines is that their squares add up to one: \(l^2 + m^2 + n^2 = 1\).

This property is intrinsic to the unit circle and the fundamental definition of trigonometric functions, which is where trigonometry ties in directly with vector algebra. Understanding the underlying trigonometry of direction cosines helps students to solve problems related to spatial orientation and line geometry with precision and ease.

Line geometry becomes particularly interesting when we examine objects in three-dimensional space. A key element to understanding 3D line geometry is grasping the concept of direction cosines, which form the bridge between algebraic expressions and geometric interpretations. When a line is described in terms of its direction cosines, it allows for a precise specification of its orientation with respect to the coordinate axes.

Furthermore, line geometry in 3D also deals with the equations of lines, planes, and the angles between them, where vector algebra plays a central role. For instance, using direction cosines can help in determining the angle between two lines, or finding the shortest distance between a point and a line. Hence, competency in 3D line geometry often relies on a robust understanding of vector algebra and trigonometry.

A solid grasp of these concepts can enhance a student's spatial reasoning, enabling them to tackle advanced topics such as finding intersections between geometric entities and solving real-world problems involving three-dimensional objects.

Trigonometry in mathematics is a study of triangles, specifically right triangles, and the relationships between their sides and the angles between those sides. It is a subject that manifests throughout various topics in mathematics, including vector algebra and line geometry. The trigonometric functions sine, cosine, and tangent are foundational, and their extensions into the unit circle provide a broader understanding that is crucial for higher mathematics.

These functions also become vital when analyzing the orientation of lines in space–direction cosines are, after all, cosines of angles. Such connections further emphasize the importance of trigonometry in solving geometric problems. It allows students to analyze and visualize mathematical concepts in a way that is applicable across a range of mathematical and physical scenarios, from the simple calculation of the sides of a right triangle to the complex interpretation of wave patterns in physics.