Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

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Direction Cosines Of A Line

If Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev are the angles which a given directed line makes with the positive directions of the axes. of x, y and z respectively, then Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev are called the direction cosines (briefly written as d.c.’s) of the line. These d.c.’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 Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev then Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev are the d.c.’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 Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev is a unit vector in the direction of that line. Clearly OP'(i.e. the line through O and parallel to BA) makes angle 180° – α, 180°– β, 180° – γ with OX, OY and OZ respectively. Hence d.c.’s of the line BA are Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev i.e., are Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev
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.’s of OP.

If l, m, n are direction cosines of any line AB, then they will satisfy Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

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.’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.’s are l, m, n. From the definition of d.r.’s. we have l/a = m/b = n/c = k (say).     Then l = ka, m = kb, n = kc. But Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev
.
Taking the positive value of k, we get l

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev
Again taking the negative value of k, we get l

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev


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 aiˆ + bjˆ + ckˆ 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 – 2nl –2lm = 0.


Sol. The given relations are l + m + n = 0 or l = –m – n ....(1)

and mn – 2nl – 2lm = 0 ...(2)
Putting the value of l from (1) in the relation (2), we get
mn – 2n (–m –n) – 2(–m – n) m = 0 or Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

To find the projection of the line joining two points Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev on the another line whose d.c.’s are l, m, n.

Let O be the origin. Then Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev
Now the unit vector along the line whose d.c.’s are  Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

∴ projection of PQ on the line whose d.c.’s are l, m, n

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

The angle θ between these two lines is given by  Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

If l1, m1, n1 and l2 , m2, n2 are two sets of real numbers, then

(l12 + m12 + n12) (l22 + m22 + n22) - (l1l2 + m1m2 + n1n2)2 = (m1n2 – m2n1)2 + (n1l2 - n2l1)2 + (l1m2 - l2m1)2

Now, we have

sin2θ = 1 - cos2θ = 1 - (l1l2 + m1m2 + n1n2)2 = (l12 + m12 + n12) (l22 + m22 + n22) - (l1l2 + m1m2 + n1n2)2

 = (m1n2 – m2n1)2 + (n1l2 - n2l1)2 + (l1m2 - l2m1)  Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Condition for perpendicularity ⇒ l1l2 + m1m2 + n1n2 = 0.

Condition for parallelism ⇒  l1 = l2, m1 = m2, n1 = n2. ⇒  Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Ex.5 Show that the lines whose d.c.’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 (–m –n) n – 5 (–m –n) m = 0 or 5m2 + 4mn – 3n2 = 0 or 5(m/n)2 + 4(m/n) – 3 = 0 ....(2)

Let l1, m1, n1 and l2, m2, n2 be the d,c,s of the two lines. Then the roots of (2) are m1/n1 and m2/n2.

product of the roots =  Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev ...(3)

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

2m (–l - m) + 3l(-l - m) - 5lm = 0  or 3(l/m)2 + 10 (l/m) + 2 = 0.

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

l1l2 + m1m2 + n1n= (2 + 3 - 5) k = 0 . k = 0.  ⇒  The lines are at right angles.

Remarks :

(a) Any three numbers a, b, c proportional to the direction cosines are called the direction ratios  Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev  same sign either +ve or –ve should be taken throughout.

Note that d.r’s of a line joining x1, y1, z1 and x2, y2, z2 are proportional to x2 – x1, y2 – y1 and z2 – z1

(b) If θ is the angle between the two lines whose d.c's are l 1 , m1, n1 and l 2 , m2, n2

cos θ = l1l2 + m1m2 + n1n2

Hence if lines are perpendicular then l1l2 + m1m2 + n1n2 = 0.

if lines are parallel then   Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

Note that if three lines are coplanar then  Direction Cosines and Direction Ratios of a Line JEE Notes | EduRev

(c) Projection of the join of two points on a line with d.c’s l, m, n are l (x2 – x1) + m(y2 – y1) + n(z2 – z1)

(d) If l1, m1, n1 and l2, m2, n2 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 l1 ± l2, m1 ± m2, n1 ± n2

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