**Coordinate System**

The three mutually perpendicular lines in a space which divides the space into eight parts and if these perpendicular lines are the coordinate axes, then it is said to be a coordinate system.

**Sign Convention**

**Distance between Two Points**

Let P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) be two given points. The distance between these points is given by

PQ √(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2} + (z_{2} – z_{1})^{2}

The distance of a point P(x, y, z) from origin O is

OP = √x^{2} + y^{2} + z^{2}

**Section Formulae**

(i) The coordinates of any point, which divides the join of points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) in the ratio m : n internally are

(mx_{2} + nx_{1} / m + n, my_{2} + ny_{1} / m + n, mz_{2} + nz_{1} / m + n)

(ii) The coordinates of any point, which divides the join of points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) in the ratio m : n externally are

(mx_{2} – nx_{1} / m – n, my_{2} – ny_{1} / m – n, mz_{2} – nz_{1} / m – n)

(iii) The coordinates of mid-point of P and Q are

(x_{1} + x_{2} / 2 , y_{1} + y_{2} / 2, z_{1} + z_{2} / 2)

(iv) Coordinates of the centroid of a triangle formed with vertices P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) and R(x_{3}, y_{3}, z_{3}) are

(x_{1} + x_{2} + x_{3} / 3 , y_{1} + y_{2} + y_{3} / 3, z_{1} + z_{2} + z_{3} / 3)

(v) **Centroid of a Tetrahedron**

If (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3}) and (x_{4}, y_{4}, z_{4}) are the vertices of a tetrahedron, then its centroid G is given by

(x_{1} + x_{2} + x_{3} + x_{4 } / 4 , y_{1} + y_{2} + y_{3} + y_{4} / 4, z_{1} + z_{2} + z_{3} + z_{4} / 4)

**Direction Cosines**

If a directed line segment OP makes angle α, β and γ with OX , OY and OZ respectively, then Cos α, cos β and cos γ are called direction cosines of up and it is represented by l, m, n.

i.e.,

l = cos α

m = cos β

and n = cos γ

If OP = r, then coordinates of OP are (lr, mr , nr)

(i) If 1, m, n are direction cosines of a vector r, then

(a) r = |r| (li + mj + nk) ⇒ r = li + mj + nk

(b) l^{2} + m^{2} + n^{2} = 1

(c) Projections of r on the coordinate axes are

(d) |r| = l|r|, m|r|, n|r| / √sum of the squares of projections of r on the coordinate axes

(ii) If P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are two points, such that the direction cosines of PQ are l, m, n. Then,

x_{2} – x_{1} = l|PQ|, y_{2} – y_{1} = m|PQ|, z_{2} – z_{1} = n|PQ|

These are projections of PQ on X , Y and Z axes, respectively.

(iii) If 1, m, n are direction cosines of a vector r and a b, c are three numbers, such that l / a = m / b = n / c.

Then, we say that the direction ratio of r are proportional to a, b, c.

Also, we have

l = a / √a^{2} + b^{2} + c^{2}, m = b / √a^{2} + b^{2} + c^{2}, n = c / √a^{2} + b^{2} + c^{2}

(iv) If θ is the angle between two lines having direction cosines l_{1}, m_{1}, n_{1} and 1_{2}, m_{2}, n_{2}, then

cos θ = l_{1}1_{2} + m_{1}m_{2} + n_{1}n_{2}

(a) Lines are parallel, if l_{1} / 1_{2} = m_{1} / m_{2} = n_{1} / n_{2}

(b) Lines are perpendicular, if l_{1}1_{2} + m_{1}m_{2} + n_{1}n_{2}

(v) If θ is the angle between two lines whose direction ratios are proportional to a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} respectively, then the angle θ between them is given by

cos θ = a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} / √a^{2}_{1} + b^{2}_{1} + c^{2}_{1} √a^{2}_{2} + b^{2}_{2} + c^{2}_{2}

Lines are parallel, if a_{1} / a_{2} = b_{1} / b_{2} = c_{1} / c_{2}

Lines are perpendicular, if a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0.

(vi) The projection of the line segment joining points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) to the line having direction cosines 1, m, n is

|(x_{2} – x_{1})l + (y_{2} – y_{1})m + (z_{2} – z_{1})n|.

(vii) The direction ratio of the line passing through points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are proportional to x_{2} – x_{1}, y_{2} – y_{1} – z_{2} – z_{1} Then, direction cosines of PQ are

x_{2} – x_{1} / |PQ|, y_{2} – y_{1} / |PQ|, z_{2} – z_{1} / |PQ|

**Area of Triangle**

If the vertices of a triangle be A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) and C(x_{3}, y_{3}, z_{3}), then

**Angle Between Two Intersecting Lines**

If l(x_{1}, m_{1}, n_{1}) and l(x_{2}, m_{2}, n_{2}) be the direction cosines of two given lines, then the angle θ between them is given by

cos θ = l_{1}1_{2} + m_{1}m_{2} + n_{1}n_{2}

(i) The angle between any two diagonals of a cube is cos^{-1} (1 / 3).

(ii) The angle between a diagonal of a cube and the diagonal of a face (of the cube is cos^{-1} (√2 / 3)

**Straight Line in Space**

The two equations of the line ax + by + cz + d = 0 and a’ x + b’ y + c’ z + d’ = 0 together represents a straight line.

1. Equation of a straight line passing through a fixed point A(x_{1}, y_{1}, z_{1}) and having direction ratios a, b, c is given by

x – x_{1} / a = y – y_{1} / b = z – z_{1} / c, it is also called the symmetrically form of a line.

Any point P on this line may be taken as (x_{1} + λa, y_{1} + λb, z_{1} + λc), where λ ∈ R is parameter. If a, b, c are replaced by direction cosines 1, m, n, then λ, represents distance of the point P from the fixed point A.

2. Equation of a straight line joining two fixed points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) is given by

x – x_{1} / x_{2} – x_{1} = y – y_{1} / y_{2} – y_{1} = z – z_{1} / z_{2} – z_{1}

3. Vector equation of a line passing through a point with position vector a and parallel to vector b is r = a + λ b, where A, is a parameter.

4. Vector equation of a line passing through two given points having position vectors a and b is r = a + λ (b – a) , where λ is a parameter.

5. (a) The length of the perpendicular from a point on the line r – a + λ b is given by

(b) The length of the perpendicular from a point P(x_{1}, y_{1}, z_{1}) on the line

where, 1, m, n are direction cosines of the line.

6. **Skew Lines** Two straight lines in space are said to be skew lines, if they are neither parallel nor intersecting.

7. **Shortest Distance** If l_{1} and l_{2} are two skew lines, then a line perpendicular to each of lines 4 and 12 is known as the line of shortest distance.

If the line of shortest distance intersects the lines l_{1} and l_{2} at P and Q respectively, then the distance PQ between points P and Q is known as the shortest distance between l_{1} and l_{2}.

8. The shortest distance between the lines

9. The shortest distance between lines r = a_{1} + λb_{1} and r = a_{2} + μb_{2} is given by

10. The shortest distance parallel lines r = a_{1} + λb_{1} and r = a_{2} + μb_{2} is given by

11. Lines r = a_{1} + λb_{1} and r = a_{2} + μb_{2} are intersecting lines, if (b_{1} * b_{2}) * (a_{2} – a_{1}) = 0.

12. The image or reflection (x, y, z) of a point (x_{1}, y_{1}, z_{1}) in a plane ax + by + cz + d = 0 is given by

x – x_{1} / a = y – y_{1} / b = z – z_{1} / c = – 2 (ax_{1} + by_{1} + cz_{1} + d) / a^{2} + b^{2} + c^{2}

13. The foot (x, y, z) of a point (x_{1}, y_{1}, z_{1}) in a plane ax + by + cz + d = 0 is given by

x – x_{1} / a = y – y_{1} / b = z – z_{1} / c = – (ax_{1} + by_{1} + cz_{1} + d) / a^{2} + b^{2} + c^{2}

14. Since, x, y and z-axes pass through the origin and have direction cosines (1, 0, 0), (0, 1, 0) and (0, 0, 1), respectively. Therefore, their equations are

x – axis : x – 0 / 1 = y – 0 / 0 = z – 0 / 0

y – axis : x – 0 / 0 = y – 0 / 1 = z – 0 / 0

z – axis : x – 0 / 0 = y – 0 / 0 = z – 0 / 1

**Plane**

A plane is a surface such that, if two points are taken on it, a straight line joining them lies wholly in the surface.

**General Equation of the Plane**

The general equation of the first degree in x, y, z always represents a plane. Hence, the general equation of the plane is ax + by + cz + d = 0. The coefficient of x, y and z in the cartesian equation of a plane are the direction ratios of normal to the plane.

**Equation of the Plane Passing Through a Fixed Point**

The equation of a plane passing through a given point (x_{1}, y_{1}, z_{1}) is given by a(x – x_{1}) + b (y — y_{1}) + c (z — z_{1}) = 0.

**Normal Form of the Equation of Plane**

(i) The equation of a plane, which is at a distance p from origin and the direction cosines of the normal from the origin to the plane are l, m, n is given by lx + my + nz = p.

(ii) The coordinates of foot of perpendicular N from the origin on the plane are (1p, mp, np).

**Intercept Form**

The intercept form of equation of plane represented in the form of

x / a + y / b + z / c = 1

where, a, b and c are intercepts on X, Y and Z-axes, respectively.

**For x intercept** Put y = 0, z = 0 in the equation of the plane and obtain the value of x. Similarly, we can determine for other intercepts.

**Equation of Planes with Given Conditions**

(i) Equation of a plane passing through the point A(x_{1}, y_{1}, z_{1}) and parallel to two given lines with direction ratios

(ii) Equation of a plane through two points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) and parallel to a line with direction ratios a, b, c is

(iii) The Equation of a plane passing through three points A(x_{1}, y_{1}, z_{1}), B(x_{2}, y_{2}, z_{2}) and C(x_{3}, y_{3}, z_{3}) is

(iv) Four points A(x_{1}, y_{1}, z_{1}), B(x_{2}, y_{2}, z_{2}), C(x_{3}, y_{3}, z_{3}) and D(x_{4}, y_{4}, z_{4}) are coplanar if and only if

(v) Equation of the plane containing two coplanar lines

**Angle between Two Planes**

The angle between two planes is defined as the angle between the normal to them from any point.

Thus, the angle between the two planes

a_{1}x + b_{1}y + c_{1}z + d_{1} = 0

and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0

is equal to the angle between the normals with direction cosines

± a_{1} / √Σ a^{2}_{1}, ± b_{1} / √Σ a^{2}_{1}, ± c_{1} / √Σ a^{2}_{1}

and ± a_{2} / √Σ a^{2}_{2}, ± b_{2} / √Σ a^{2}_{2}, ± c_{2} / √Σ a^{2}_{2}

If θ is the angle between the normals, then

cos θ = ± a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} / √a^{2}_{1} + b^{2}_{1} + c^{2}_{1} √a^{2}_{2} + b^{2}_{2} + c^{2}_{2}

**Parallelism and Perpendicularity of Two Planes**

Two planes are parallel or perpendicular according as the normals to them are parallel or perpendicular.

Hence, the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0

are parallel, if a_{1} / a_{2} = b_{1} / b_{2} = c_{1} / c_{2} and perpendicular, if a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0.

Note The equation of plane parallel to a given plane ax + by + cz + d = 0 is given by ax + by + cz + k = 0, where k may be determined from given conditions.

**Angle between a Line and a Plane**

**In Vector Form** The angle between a line r = a + λ b and plane r *• n = d, is defined as the complement of the angle between the line and normal to the plane:

sin θ = n * b / |n||b|

**In Cartesian Form** The angle between a line x – x_{1} / a_{1} = y – y_{1} / b_{1} = z – z_{1} / c_{1}

and plane a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 is sin θ = a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} / √a^{2}_{1} + b^{2}_{1} + c^{2}_{1} √a^{2}_{2} + b^{2}_{2} + c^{2}_{2}

**Distance of a Point from a Plane**

Let the plane in the general form be ax + by + cz + d = 0. The distance of the point P(x_{1}, y_{1}, z_{1}) from the plane is equal to

If the plane is given in, normal form lx + my + nz = p. Then, the distance of the point P(x_{1}, y_{1}, z_{1}) from the plane is |lx_{1} + my_{1} + nz_{1} – p|.

**Distance between Two Parallel Planes**

If ax + by + cz + d_{1} = 0 and ax + by + cz + d_{2} = 0 be equation of two parallel planes. Then, the distance between them is

**Bisectors of Angles between Two Planes**

The bisector planes of the angles between the planes

a_{1}x + b_{1}y + c_{1}z + d_{1} = 0, a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 is

a_{1}x + b_{1}y + c_{1}z + d_{1} / √Σa^{2}_{1} = ± a_{2}x + b_{2}y + c_{2}z + d_{2} / √Σa^{2}_{2}

One of these planes will bisect the acute angle and the other obtuse angle between the given plane.

**Sphere**

A sphere is the locus of a point which moves in a space in such a way that its distance from a fixed point always remains constant.

**General Equation of the Sphere**

**In Cartesian Form** The equation of the sphere with centre (a, b, c) and radius r is

(x – a)^{2} + (y – b)^{2} + (z – c)^{2} = r^{2} …….(i)

In generally, we can write

x^{2} + y^{2} + z^{2} + 2ux + 2vy + 2wz + d = 0

Here, its centre is (-u, v, w) and radius = √u^{2} + v^{2} + w^{2} – d

**In Vector Form** The vector equation of a sphere of radius a and Centre having position vector c is |r – c| = a

**Important Points to be Remembered**

(i) The general equation of second degree in x, y, z is ax^{2} + by^{2} + cz^{2} + 2hxy + 2kyz + 2lzx + 2ux + 2vy + 2wz + d = 0

represents a sphere, if

(a) a = b = c (≠ 0)

(b) h = k = 1 = 0

The equation becomes

ax^{2} + ay^{2} + az^{2} + 2ux + 2vy + 2wz + d – 0 …(A)

To find its centre and radius first we make the coefficients of x^{2}, y^{2} and z^{2} each unity by dividing throughout by a.

Thus, we have

x^{2}+y^{2}+z^{2} + (2u / a) x + (2v / a) y + (2w / a) z + d / a = 0 …..(B)

∴ Centre is (- u / a, – v / a, – w / a)

and radius = √u^{2} / a^{2} + v^{2} / a^{2} + w^{2} / a^{2} – d / a

= √u^{2} + v^{2} + w^{2} – ad / |a| .

(ii) Any sphere concentric with the sphere

x^{2} + y^{2} + z^{2} + 2ux + 2vy + 2wz + d = 0

is x^{2} + y^{2} + z^{2} + 2ux + 2vy + 2wz + k = 0

(iii) Since, r^{2} = u^{2} + v^{2} + w^{2} — d, therefore, the Eq. (B) represents a real sphere, if u^{2} +v^{2} + w^{2} — d > 0

(iv) The equation of a sphere on the line joining two points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) as a diameter is

(x – x_{1}) (x – x_{1}) + (y – y_{1}) (y – y_{2}) + (z – z_{1}) (z – z_{2}) = 0.

(v) The equation of a sphere passing through four non-coplanar points (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3}) and (x_{4}, y_{4}, z_{4}) is

**Tangency of a Plane to a Sphere**

The plane lx + my + nz = p will touch the sphere x^{2} + y^{2} + z^{2} + 2ux + 2vy + 2 wz + d = 0, if length of the perpendicular from the centre ( – u, – v,— w)= radius,

i.e., |lu – mv – nw – p| / √l^{2} + m^{2} + n^{2} = √u^{2} + v^{2} + w^{2} – d

(lu – mv – nw – p)^{2} = (u^{2} + v^{2} + w^{2} – d) (l^{2} + m^{2} + n^{2})

**Plane Section of a Sphere**

Consider a sphere intersected by a plane. The set of points common to both sphere and plane is called a plane section of a sphere.

In ΔCNP, NP^{2} = CP^{2} – CN^{2} = r^{2} – p^{2}

∴ NP = √r^{2} – p^{2}

Hence, the locus of P is a circle whose centre is at the point N, the foot of the perpendicular from the centre of the sphere to the plane.

The section of sphere by a plane through its centre is called a great circle. The centre and radius of a great circle are the same as those of the sphere.

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