Maharashtra Board Class 12 Maths Part 1 Chapter 6 Line And Plane PDF Download

Line And Plane

Let's Study

6.1 Vector and Cartesian equations of a line.

6.1.1 Passing through a point and parallel to a vector.

6.1.2 Passing through two points.

6:2 Distance of a point from a line.

6.3 Skew lines

6.3.1 Distance between skew lines

6.3.2 Distance between parallel lines.

6.4 Equations of Plane:

6.4.1 Passing through a point and perpendicular to a vector.

6.4.2 Passing through a point and parallel to two vectors.

6.4.3 Passing through three non-collinear points.

6.4.4 In normal form.

6.4.5 Passing through the intersection of two planes.

6.5 Angle between planes:

6.5.1 Angle between two planes.

6.5.2 Angle between a line and a plane.

6.6 Coplanarity of two lines.

6.7 Distance of a point from a plane.

Let's Recall

A line in space is completely determined by a point on it and its direction. Two points on a line determine the direction of the line. Let us derive equations of lines in different forms and discuss parallel lines.

6.1 Vector And Cartesian Equations Of A Line

Line in space is a locus. Points on line have position vectors. Position vector of a point determines the position of the point in space. In this topic position vector of a variable point on line will be denoted by \(\vec{r}\).

Teacher's Note

A line in space needs two things to be fully described: a point it passes through and a direction it goes in. Think of a train track—you know which station it passes through, and you know which way it goes.

Exam Trick

Remember: To write a line equation, you always need a point (starting place) and a direction (which way to go). It is like giving someone your house address and then telling them which road to take.

Points To Remember

A line needs a point and a direction vector to be completely described.

The position vector shows where a point is in space.

Two points on a line can tell us the direction of that line.

Lines can be written in vector form or Cartesian form.

6.1.1 Equation Of A Line Passing Through A Given Point And Parallel To Given Vector

Theorem 6.1

The vector equation of the line passing through \(A(\vec{a})\) and parallel to vector \(\vec{b}\) is \(\vec{r} = \vec{a} + \lambda\vec{b}\).

Proof

Let \(L\) be the line which passes through \(A(\vec{a})\) and parallel to vector \(\vec{b}\).

Let \(P(\vec{r})\) be a variable point on the line \(L\).

\(\therefore \overrightarrow{AP}\) is parallel to \(\vec{b}\).

\(\therefore \overrightarrow{AP} = \lambda\vec{b}\), where \(\lambda\) is a scalar.

\(\therefore \overrightarrow{OP} - \overrightarrow{OA} = \lambda\vec{b}\)

\(\therefore \vec{r} - \vec{a} = \lambda\vec{b}\)

\(\therefore \vec{r} = \vec{a} + \lambda\vec{b}\)

This is the required vector equation of the line.

Each real value of \(\lambda\) corresponds to a point on line \(L\) and conversely each point on \(L\) determines unique value of \(\lambda\). There is one to one correspondence between points on \(L\) and values of \(\lambda\). Here \(\lambda\) is called a parameter and equation \(\vec{r} = \vec{a} + \lambda\vec{b}\) is called the parametric form of vector equation of line.

Activity: Write position vectors of any three points on the line \(\vec{r} = \vec{a} + \lambda\vec{b}\).

Remark: The equation of line passing through \(A(\vec{a})\) and parallel to vector \(\vec{b}\) can also be expressed as \((\vec{r} - \vec{a}) \times \vec{b} = \vec{0}\). This equation is called the non-parametric form of vector equation of line.

Theorem 6.2

The Cartesian equations of the line passing through \(A(x_1, y_1, z_1)\) and having direction ratios \(a, b, c\) are \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\).

Proof

Let \(L\) be the line which passes through \(A(x_1, y_1, z_1)\) and has direction ratios \(a, b, c\).

Let \(P(x, y, z)\) be a variable point on the line \(L\) other than \(A\).

\(\therefore\) Direction ratios of \(L\) are \(x - x_1\), \(y - y_1\), \(z - z_1\).

But direction ratios of line \(L\) are \(a, b, c\).

\(\therefore \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\) are the required Cartesian equations of the line.

In Cartesian form line cannot be represented by a single equation.

Remark

If \(\vec{b} = a_1\vec{i} + b_1\vec{j} + c_1\vec{k}\) then \(a_1, b_1, c_1\) are direction ratios of the line and conversely if \(a_1, b_1, c_1\) are direction ratios of a line then \(\vec{b} = a_1\vec{i} + b_1\vec{j} + c_1\vec{k}\) is parallel to the line.

The equations \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} = \lambda\) are called the symmetric form of Cartesian equations of line.

The equations \(x = x_1 + \lambda a\), \(y = y_1 + \lambda b\), \(z = z_1 + \lambda c\) are called parametric form of the Cartesian equations of line.

Teacher's Note

Direction ratios are like the slope of a line. They tell us how much the line moves in each direction. Just like a road goes so many kilometers east and so many kilometers north, a line in space goes in three directions.

Exam Trick

Remember: In the equation \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\), the numbers \(a\), \(b\), \(c\) are direction ratios. They are like the "speed" in each direction.

Points To Remember

Direction ratios tell us how a line is tilted in space.

Direction cosines are direction ratios divided by their total length.

The parameter \(\lambda\) helps us find any point on the line.

Three equations give us one line in space.

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