Three-Dimensional Geometry – Line | Equation Of A Line

ANGLE BETWEEN TWO LINES | 3-DIMENSIONAL GEOMETRY | NCERT CLASS 12 MATHS

In accordance with NCERT Class 12 Maths, 3D geometry alludes to the mathematics of shapes in three-dimensional space and comprises of 3 coordinates. These 3 coordinates are x-coordinate, y-coordinate, and z-coordinate. In three-dimensional space, there is a necessity of three parameters to locate the specific area of a point. Dimension, in like manner speech, signifies the proportion of an item’s size, for example, a box, generally given as height, length, and breadth. In geometry, the thought of dimension is an augmentation of the possibility that a line speaks to one-dimensional, a plane happens to be two-dimensional, and space is three-dimensional.

The arrangement of a three-dimensional Cartesian coordinate system is referred to as the origin just as a premise including three mutually perpendicular vectors with respect to NCERT Class 12 Maths. These vectors appropriately clarify the three coordinate axes which are: the x-, y-, and z-axis. Specialists additionally call them as abscissa, ordinate and applicate pivot, separately.

ANGLE BETWEEN TWO LINES

According to NCERT Class 12 Maths, Angle between two lines alludes to the angle between two intersecting lines. This is because the angle between the two perpendicular lines is and that angle between two parallel lines will be. Thus, we will presently take a gander at how the angle between two lines is determined.

CARTESIAN FORM

Let and be two lines passing through the origin and with direction ratios and, respectively. Let P be a point on and Q be a point on. Consider the directed lines OP and OQ as given in the following figure. Let θ be the acute angle between OP and OQ. Now recall that the directed line segments OP and OQ are vectors with components and, respectively. Therefore, the angle between two lines formula is given by:

ANGLE BETWEEN TWO LINES FORMULA:

Here, equations of the two lines are of form:

VECTOR FORM

Let the equations of two lines be

denotes angle between the two lines.

Then, Angle between two lines formula will be:

VECTOR AND CARTESIAN EQUATIONS OF A LINE

1.

In Cartesian Form:

Let the coordinates of the given point A (x₁ , y₁, z₁ ) be and the direction ratios of the line are

a, b, c. Consider the coordinates of any point P be (x, y, z). Then the Cartesian Equation of a line is

2. Equation of a line passing through two given points:

Let a and b be the position vectors of two points A (x₁ , y₁, z₁ )and , B (x₁ , y₁, z₁ ) respectively that are lying on a line.

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IMPORTANT FORMULAE OF 3-DIMENSIONAL GEOMETRY WITH RESPECT TO NCERT CLASS 12 MATHS

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