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**Direction Cosines Of A Line**

If are the angles which a given directed line makes with the positive directions of the axes. of x, y and z respectively, then are called the direction cosines (briefly written as d.c.â€™s) of the line. These d.c.â€™s are usually denote by l, m, n.

Let AB be a given line. Draw a line OP parallel to the line AB and passing through the origin O. Measure angles then are the d.c.â€™s of the line AB. It can be easily seen that l, m, n, are the direction cosines of a line if and only if is a unit vector in the direction of that line. Clearly OP'(i.e. the line through O and parallel to BA) makes angle 180Â° â€“ Î±, 180Â°â€“ Î², 180Â° â€“ Î³ with OX, OY and OZ respectively. Hence d.c.â€™s of the line BA are i.e., are

If the length of a line OP through the origin O be r, then the co-ordinates of P are (lr, mr, nr) where l, m, n are the d c.â€™s of OP.

If l, m, n are direction cosines of any line AB, then they will satisfy

**Direction Ratios :**

If the direction cosines l, m, n of a given line be proportional to any three numbers a, b, c respectively, then the numbers a, b, c are called direction ratios (briefly written as d.r.â€™s of the given line.

**Relation Between Direction Cosines And Direction Ratios :**

Let a, b, c be the direction ratios of a line whose d.c.â€™s are l, m, n. From the definition of d.r.â€™s. we have l/a = m/b = n/c = k (say). Then l = ka, m = kb, n = kc. But

.

Taking the positive value of k, we get l

Again taking the negative value of k, we get l

**Remark**. Direction cosines of a line are unique. But the direction ratios of a line are by no means unique. If a, b, c are direction ratios of a line, then ka, kb, kc are also direction ratios of that line where k is any non-zero real number. Moreover if a, b, c are direction ratios of a line, then aiË† + bjË† + ckË† is a vector parallel to that line.

**Ex.4 Find the direction cosines l + m + n of the two lines which are connected by the relation l + m + n = 0 and mn â€“ 2nl â€“2lm = 0.**

**Sol.** The given relations are l + m + n = 0 or l = â€“m â€“ n ....(1)

and mn â€“ 2nl â€“ 2lm = 0 ...(2)

Putting the value of l from (1) in the relation (2), we get

mn â€“ 2n (â€“m â€“n) â€“ 2(â€“m â€“ n) m = 0 or

**To find the projection of the line joining two points **** on the another line whose d.c.â€™s are l, m, n.**

Let O be the origin. Then

Now the unit vector along the line whose d.c.â€™s are

âˆ´ projection of PQ on the line whose d.c.â€™s are l, m, n

The angle Î¸ between these two lines is given by

If l_{1}, m_{1}, n_{1} and l_{2} , m_{2}, n_{2} are two sets of real numbers, then

(l_{1}^{2} + m_{1}^{2} + n_{1}^{2}) (l_{2}^{2} + m_{2}^{2} + n_{2}^{2}) - (l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2})^{2} = (m_{1}n_{2} â€“ m_{2}n_{1})^{2} + (n_{1}l_{2} - n_{2}l_{1})^{2} + (l_{1}m_{2} - l_{2}m_{1})^{2}

Now, we have

sin2Î¸ = 1 - cos2Î¸ = 1 - (l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2})^{2} = (l_{1}^{2} + m_{1}^{2} + n_{1}^{2}) (l_{2}^{2} + m_{2}^{2} + n_{2}^{2}) - (l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2})^{2}

= (m_{1}n_{2} â€“ m_{2}n_{1})^{2} + (n_{1}l_{2} - n_{2}l_{1})^{2} + (l_{1}m_{2} - l_{2}m_{1})^{2 }

Condition for perpendicularity â‡’ l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0.

Condition for parallelism â‡’ l_{1} = l_{2}, m_{1} = m_{2}, n_{1} = n_{2}. â‡’

**Ex.5 Show that the lines whose d.c.â€™s are given by l + m + n = 0 and 2mn + 3ln - 5lm = 0 are at right angles.**

Sol. From the first relation, we have l = -m - n....(1)

Putting this value of l in the second relation, we have

2mn + 3 (â€“m â€“n) n â€“ 5 (â€“m â€“n) m = 0 or 5m^{2} + 4mn â€“ 3n^{2} = 0 or 5(m/n)^{2} + 4(m/n) â€“ 3 = 0 ....(2)

Let l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} be the d,c,s of the two lines. Then the roots of (2) are m_{1}/n_{1} and m_{2}/n_{2}.

product of the roots = ...(3)

Again from (1), n = â€“ l - m and putting this value of n in the second given relation, we have

2m (â€“l - m) + 3l(-l - m) - 5lm = 0 or 3(l/m)^{2} + 10 (l/m) + 2 = 0.

l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2 }= (2 + 3 - 5) k = 0 . k = 0. â‡’ The lines are at right angles.

**Remarks :**

(a) Any three numbers a, b, c proportional to the direction cosines are called the direction ratios same sign either +ve or â€“ve should be taken throughout.

Note that d.râ€™s of a line joining x_{1}, y_{1}, z_{1} and x_{2}, y_{2}, z_{2} are proportional to x_{2} â€“ x_{1}, y_{2} â€“ y_{1} and z_{2} â€“ z_{1}

(b) If Î¸ is the angle between the two lines whose d.c's are l_{ 1} , m_{1}, n_{1} and l_{ 2} , m_{2}, n_{2}

cos Î¸ = l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}

Hence if lines are perpendicular then l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0.

if lines are parallel then

Note that if three lines are coplanar then

(c) Projection of the join of two points on a line with d.câ€™s l, m, n are l (x_{2} â€“ x_{1}) + m(y_{2} â€“ y_{1}) + n(z_{2} â€“ z_{1})

(d) If l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} are the d.c.'s of two concurrent lines, show that the d.c's of two lines bisecting the angles between them are proportional to l_{1} Â± l_{2}, m_{1} Â± m_{2}, n_{1} Â± n_{2}

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