**CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry**

CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry:-In Class XI, while studying Analytical Geometry in two

CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometrydimensions, and the introduction to three dimensional geometry, we confined to the Cartesian methods only. In the previous chapter of this book, we have studied some basic concepts of vectors. We will now use vector algebra to three dimensional geometry. The purpose of this approach to 3-dimensional geometry is that it makes the study simple and elegant*.

In this chapter, we shall study the direction cosines and direction ratios of a line joining two points and also discuss about the equations of lines and planes in space under different conditions, angle between two lines, two planes, a line and a plane, shortest distance between two skew lines and distance of a point from a plane. Most of the above results are obtained in vector form. Nevertheless, we shall also translate these results in the Cartesian form which, at times, presents a more clear geometric and analytic picture of the situation.

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**CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry:-****Direction Cosines and Direction Ratios of a Line**

From Chapter 10, recall that if a directed line L passing through the origin makes angles α, β and γ with x, y and z-axes, respectively, called direction angles, then cosine of these angles, namely, cos α, cos β and cos γ are called direction cosines of the directed line L. If we reverse the direction of L, then the direction angles are replaced by their supplements, i.e., π −α , π − β and π − γ . Thus, the signs of the direction cosines are reversed.

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**CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry:-****Equation of a Line in Space**

We have studied equation of lines in two dimensions in Class XI, we shall now study the vector and cartesian equations of a line in space. A line is uniquely determined if

(i) it passes through a given point and has given direction, or

(ii) it passes through two given points.

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**CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry:-****Derivation of cartesian form from vector form**

Let the coordinates of the given point A be (x1, y1, z1) and the direction ratios of the line be a, b, c. Consider the coordinates of any point P be (x, y, z). Then r = xiˆ + yˆj + zkˆ ; a x iˆ y ˆj z kˆ 1 1 1 = + + and b = a iˆ + b ˆj + c kˆ

Substituting these values in (1) and equating the coefficients of iˆ, ˆj and kˆ , we get

x = x1 + λ a; y = y1 + λ b; z = z1+ λ c … (2)

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**CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry:-****Shortest Distance between Two Lines**

If two lines in space intersect at a point, then the shortest distance between them is zero. Also, if two lines in space are parallel, then the shortest distance between them will be the perpendicular distance, i.e. the length of the perpendicular drawn from a point on one line onto the other line. Further, in a space, there are lines which are neither intersecting nor parallel. In fact, such pair of lines are non coplanar and are called skew lines.

The line GE that goes diagonally across the ceiling and the line DB passes through one corner of the ceiling directly above A and goes diagonally down the wall. These lines are skew because they are not parallel and also never meet. By the shortest distance between two lines we mean the join of a point in one line with one point on the other line so that the length of the segment so obtained is the smallest. For skew lines, the line of the shortest distance will be perpendicular to both the lines.

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**CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry:-****Plane**

A plane is determined uniquely if any one of the following is known:

(i) the normal to the plane and its distance from the origin is given, i.e., equation of a plane in normal form.

(ii) it passes through a point and is perpendicular to a given direction.

(iii) it passes through three given non collinear points.

Now we shall find vector and Cartesian equations of the planes.

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**CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry:-****Cartesian form**

Equation (2) gives the vector equation of a plane, where nˆ is the unit vector normal to the plane. Let P(x, y, z) be any point on the plane. Then

OP = r = x iˆ + y ˆj + z kˆ

Let l, m, n be the direction cosines of nˆ . Then

nˆ = l iˆ + m ˆj + n kˆ

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**CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry:-****Angle between Two Planes**

**Definition 2** The angle between two planes is defined as the angle between their normals (Fig 11.18 (a)). Observe that if θ is an angle between the two planes, then so is 180 – θ (Fig 11.18 (b)). We shall take the acute angle as the angles between two planes.

**CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry:-Angle between a Line and a Plane**

**Definition 3** The angle between a line and a plane is the complement of the angle between the line and normal to the plane (Fig 11.20). Vector form If the equation of the line is r a b = + λ and the equation of the plane is r ⋅ n = d . Then the angle θ between the line and the normal to the plane is

**Summary**

- Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.
- If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1.
- Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are

2 1 , 2 1 , 2 1PQ PQ PQx − x y − y z − z

where PQ = ( )22 122 122 1 (x − x ) + ( y − y ) + z − z

- Direction ratios of a line are the numbers which are proportional to the direction cosines of a line.
- If l, m, n are the direction cosines and a, b, c are the direction ratios of a line
- Skew lines are lines in space which are neither parallel nor intersecting. They lie in different planes.
- Angle between skew lines is the angle between two intersecting lines drawn from any point (preferably through the origin) parallel to each of the

skew lines. - Vector equation of a line that passes through the given point whose position vector is a and parallel to a given vector b is r a b = +λ .
- Shortest distance between two skew lines is the line segment perpendicular to both the lines.
- In the vector form, equation of a plane which is at a distance d from the origin, and nˆ is the unit vector normal to the plane through the origin is

r ⋅ nˆ = d . - Equation of a plane which is at a distance of d from the origin and the direction cosines of the normal to the plane as l, m, n is lx + my + nz = d.
- The equation of a plane through a point whose position vector is a and perpendicular to the vector Nis ( r − a ) . N = 0 .
- Equation of a plane passing through three non collinear points (x1, y1, z1),
- Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point (x1, y1, z1) is

A (x – x1) + B (y – y1) + C (z – z1 ) = 0

### CBSE Class 12 Commerce Mathematics Unit IV Vectors and Three Dimensional Geometry

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