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**Directional angles:** The angles made by a line with the positive directions of the X, Y and Z axes are called directional angles.

**Directional cosines:** If Î±,ÃŸ and Î“ are the directional angles of a directed line L, then cos Î±, cos Î² and cos Î³ are called the directional cosines of directed line L.

**By convention**:

cos Î± = l

cos Î² = m

cos Î³ = n

Thus, the directional cosines of l are:

cos (Ï€-Î±) = - cos Î±

cos (Ï€-Î²) = - cos ÃŸ

cos (Ï€-Î³) = - cos Î“

If L_{1}âˆ¥ L_{2}:

Direction cosines of L_{1 }= Direction cosines of L_{2}

Relation between direction cosines of a line:

Let P = (x,y,z)

OP = r

â‡’ âˆš((x - 0)^{2} + (y - 0)^{2 }+ (z - 0)^{2}) = r

â‡’ âˆš(x^{2} + y^{2 }+ z^{2}) = r

â‡’ x^{2} + y^{2 }+ z^{2} = r^{2} ...... (1)

âˆ´ OA = x

OB = *y*

OC = *z*

Let Î±,ÃŸ and Î“ be the angles made by line L_{1} with the X, Y and Z axes, respectively.

âˆ´ Direction cosines of line L are cos Î± ,cos Î² , cos Î³

In right-angled âˆ†OAP:

cos Î± = l = OA/OP = x/r

â‡’ *x* *=* lr ...(2)

In right-angled âˆ†OBP:

cos Î² = m = OB/OP = y/r

â‡’ *y* *=* mr ...(3)

In right-angled âˆ†OCP:

cos Î³= n = OC/OP = z/r

â‡’ *z* *=* nr ...(4)

From equations (2), (3) and (4):

x^{2} + y^{2 }+ z^{2 }= l^{2} r^{2}+ m^{2} r^{2} + n^{2} r^{2}

â‡’ x^{2} + y^{2 }+ z^{2} = r^{2}(l^{2} + m^{2} + n^{2})

â‡’ l^{2} + m^{2} + n^{2} = (x^{2} + y^{2 }+ z^{2})/r^{2} â€¦(5)

From equations (1) and (5):

l^{2} + m^{2} + n^{2} = (x^{2} + y^{2 }+ z^{2})/(âˆš(x^{2} + y^{2 }+ z^{2}))^{2}

â‡’ l^{2} + m^{2} + n^{2} = (x^{2} + y^{2 }+ z^{2})/(x^{2} + y^{2 }+ z^{2})

â‡’ l^{2} + m^{2} + n^{2} = 1

Direction Ratios

*x* *=* lr

*y* *=* mr

*z* *=* nr

Direction ratios: Any three numbers that are proportional to the direction cosines of a line are called the direction ratios of the line.

If l,m,n are the direction cosines of a line, then a,b,c are its direction ratios such that:

*l* *=* aÎ»

*m* *=* bÎ»

*n* *=* cÎ»

where Î» â‰ 0 and Î» âˆˆ R

Relation between direction cosines and direction ratios:

If l, m,n are the direction cosines of a line and a,b,c form a set of its direction ratios, then:

l/a = m/b = n/c = *k* (constant)

â‡’ l = ak ...(1)

m = bk ...(2)

n = ck ...(3)

From equations (1), (2) and (3):

l^{2} + m^{2} + n^{2} = (a^{2}k^{2} + b^{2}k^{2 }+ c^{2}k^{2})

â‡’ l^{2} + m^{2} + n^{2} = k^{2 }(a^{2} + b^{2}^{ }+ c^{2}) â€¦ (4)

We know that l^{2} + m^{2} + n^{2} = 1. Thus, in equation (4):

k^{2 }(a^{2} + b^{2}^{ }+ c^{2}) = 1

â‡’ k^{2} = 1/(a^{2} + b^{2}^{ }+ c^{2})

â‡’ k = Â± 1/âˆš(a^{2} + b^{2}^{ }+ c^{2}) â€¦ (5)

From equation (1), (2), (3) and (5):

l = a (Â± 1/âˆš(a^{2} + b^{2}^{ }+ c^{2}))= Â± a/âˆš(a^{2} + b^{2}^{ }+ c^{2})

m = b (Â± 1/âˆš(a^{2} + b^{2}^{ }+ c^{2}))= Â± b/âˆš(a^{2} + b^{2}^{ }+ c^{2})

n = c (Â± 1/âˆš(a^{2} + b^{2}^{ }+ c^{2})) =Â± c/âˆš(a^{2} + b^{2}^{ }+ c^{2})

l = Â± a/âˆš(a^{2} + b^{2}^{ }+ c^{2})

m = Â± b/âˆš(a^{2} + b^{2}^{ }+ c^{2})

n = Â± c/âˆš(a^{2} + b^{2}^{ }+ c^{2})

Direction Cosines of a Line Passing through Two Given Points

Direction cosines of a line passing through two given points:

Let A= (x_{1},y_{1},z_{1}) B = (x_{2},y_{2},z_{2})

Direction cosines of given line = cos Î±, cos Î², cos Î³

In right-angled âˆ†BAC:

cos Î² = AC/AB â€¦(1)

AC = y_{2} - y_{1}

AB = âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

â‡’ cos Î² = (y_{2} - y_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

Similarly:

cos Î± = (x_{2} - x_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

cos Î³ = (z_{2} - z_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

Direction cosines of a line passing through (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{3}):

cos Î± (l) = (x_{2} - x_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

cos Î² (m) = (y_{2} - y_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

cos Î³ (n) = (z_{2} - z_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

Direction ratios of a line passing through (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{3}):

(x_{2} - x_{1}), (y_{2} - y_{1}), (z_{2} - z_{1}) and (x_{1} - x_{2}), (y_{1} - y_{2}), (z_{1} - z_{2})

**Directional angles:** The angles made by a line with the positive directions of the X, Y and Z axes are called directional angles.

**Directional cosines:** If Î±,ÃŸ and Î“ are the directional angles of a directed line L, then cos Î±, cos Î² and cos Î³ are called the directional cosines of directed line L.

**By convention**:

cos Î± = l

cos Î² = m

cos Î³ = n

Thus, the directional cosines of l are:

cos (Ï€-Î±) = - cos Î±

cos (Ï€-Î²) = - cos ÃŸ

cos (Ï€-Î³) = - cos Î“

If L_{1}âˆ¥ L_{2}:

Direction cosines of L_{1 }= Direction cosines of L_{2}

Relation between direction cosines of a line:

Let P = (x,y,z)

OP = r

â‡’ âˆš((x - 0)^{2} + (y - 0)^{2 }+ (z - 0)^{2}) = r

â‡’ âˆš(x^{2} + y^{2 }+ z^{2}) = r

â‡’ x^{2} + y^{2 }+ z^{2} = r^{2} ...... (1)

âˆ´ OA = x

OB = *y*

OC = *z*

Let Î±,ÃŸ and Î“ be the angles made by line L_{1} with the X, Y and Z axes, respectively.

âˆ´ Direction cosines of line L are cos Î± ,cos Î² , cos Î³

In right-angled âˆ†OAP:

cos Î± = l = OA/OP = x/r

â‡’ *x* *=* lr ...(2)

In right-angled âˆ†OBP:

cos Î² = m = OB/OP = y/r

â‡’ *y* *=* mr ...(3)

In right-angled âˆ†OCP:

cos Î³= n = OC/OP = z/r

â‡’ *z* *=* nr ...(4)

From equations (2), (3) and (4):

x^{2} + y^{2 }+ z^{2 }= l^{2} r^{2}+ m^{2} r^{2} + n^{2} r^{2}

â‡’ x^{2} + y^{2 }+ z^{2} = r^{2}(l^{2} + m^{2} + n^{2})

â‡’ l^{2} + m^{2} + n^{2} = (x^{2} + y^{2 }+ z^{2})/r^{2} â€¦(5)

From equations (1) and (5):

l^{2} + m^{2} + n^{2} = (x^{2} + y^{2 }+ z^{2})/(âˆš(x^{2} + y^{2 }+ z^{2}))^{2}

â‡’ l^{2} + m^{2} + n^{2} = (x^{2} + y^{2 }+ z^{2})/(x^{2} + y^{2 }+ z^{2})

â‡’ l^{2} + m^{2} + n^{2} = 1

Direction Ratios

*x* *=* lr

*y* *=* mr

*z* *=* nr

Direction ratios: Any three numbers that are proportional to the direction cosines of a line are called the direction ratios of the line.

If l,m,n are the direction cosines of a line, then a,b,c are its direction ratios such that:

*l* *=* aÎ»

*m* *=* bÎ»

*n* *=* cÎ»

where Î» â‰ 0 and Î» âˆˆ R

Relation between direction cosines and direction ratios:

If l, m,n are the direction cosines of a line and a,b,c form a set of its direction ratios, then:

l/a = m/b = n/c = *k* (constant)

â‡’ l = ak ...(1)

m = bk ...(2)

n = ck ...(3)

From equations (1), (2) and (3):

l^{2} + m^{2} + n^{2} = (a^{2}k^{2} + b^{2}k^{2 }+ c^{2}k^{2})

â‡’ l^{2} + m^{2} + n^{2} = k^{2 }(a^{2} + b^{2}^{ }+ c^{2}) â€¦ (4)

We know that l^{2} + m^{2} + n^{2} = 1. Thus, in equation (4):

k^{2 }(a^{2} + b^{2}^{ }+ c^{2}) = 1

â‡’ k^{2} = 1/(a^{2} + b^{2}^{ }+ c^{2})

â‡’ k = Â± 1/âˆš(a^{2} + b^{2}^{ }+ c^{2}) â€¦ (5)

From equation (1), (2), (3) and (5):

l = a (Â± 1/âˆš(a^{2} + b^{2}^{ }+ c^{2}))= Â± a/âˆš(a^{2} + b^{2}^{ }+ c^{2})

m = b (Â± 1/âˆš(a^{2} + b^{2}^{ }+ c^{2}))= Â± b/âˆš(a^{2} + b^{2}^{ }+ c^{2})

n = c (Â± 1/âˆš(a^{2} + b^{2}^{ }+ c^{2})) =Â± c/âˆš(a^{2} + b^{2}^{ }+ c^{2})

l = Â± a/âˆš(a^{2} + b^{2}^{ }+ c^{2})

m = Â± b/âˆš(a^{2} + b^{2}^{ }+ c^{2})

n = Â± c/âˆš(a^{2} + b^{2}^{ }+ c^{2})

Direction Cosines of a Line Passing through Two Given Points

Direction cosines of a line passing through two given points:

Let A= (x_{1},y_{1},z_{1}) B = (x_{2},y_{2},z_{2})

Direction cosines of given line = cos Î±, cos Î², cos Î³

In right-angled âˆ†BAC:

cos Î² = AC/AB â€¦(1)

AC = y_{2} - y_{1}

AB = âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

â‡’ cos Î² = (y_{2} - y_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

Similarly:

cos Î± = (x_{2} - x_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

cos Î³ = (z_{2} - z_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

Direction cosines of a line passing through (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{3}):

cos Î± (l) = (x_{2} - x_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

cos Î² (m) = (y_{2} - y_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

cos Î³ (n) = (z_{2} - z_{1})/âˆš((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})

Direction ratios of a line passing through (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{3}):

(x_{2} - x_{1}), (y_{2} - y_{1}), (z_{2} - z_{1}) and (x_{1} - x_{2}), (y_{1} - y_{2}), (z_{1} - z_{2})