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Direction Cosines Of A Line

If Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced 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 | Mathematics (Maths) for JEE Main & Advanced 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 | Mathematics (Maths) for JEE Main & Advanced then Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced 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 | Mathematics (Maths) for JEE Main & Advanced 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 | Mathematics (Maths) for JEE Main & Advanced i.e., are Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced
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 | Mathematics (Maths) for JEE Main & Advanced

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 | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced
.
Taking the positive value of k, we get l

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced
Again taking the negative value of k, we get l

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced


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.

Example: 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 | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Finding Projection of a line

To find the projection of the line joining two points Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced 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 | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced
Now the unit vector along the line whose d.c.’s are  Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

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

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

The angle θ between these two lines is given by  Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

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 | Mathematics (Maths) for JEE Main & Advanced

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

Condition for parallelism ⇒  l1 = l2, m1 = m2, n1 = n2. ⇒  Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Example: 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 | Mathematics (Maths) for JEE Main & Advanced ...(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 | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

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 | Mathematics (Maths) for JEE Main & Advanced  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 | Mathematics (Maths) for JEE Main & Advanced

Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

Note that if three lines are coplanar then  Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced

(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

The document Direction Cosines and Direction Ratios of a Line | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on Direction Cosines and Direction Ratios of a Line - Mathematics (Maths) for JEE Main & Advanced

1. What are direction cosines and direction ratios of a line?
Ans. Direction cosines of a line are the cosines of the angles that the line makes with the positive directions of the coordinate axes. Direction ratios of a line are the ratios of the direction cosines of the line.
2. How can I find the direction cosines of a line?
Ans. To find the direction cosines of a line, first, determine the angles that the line makes with the positive directions of the coordinate axes. Then, take the cosines of these angles to obtain the direction cosines.
3. Can you explain how to calculate the direction ratios of a line?
Ans. Yes, to calculate the direction ratios of a line, divide each of the direction cosines of the line by the greatest common factor (GCF) of the direction cosines. This will give you the direction ratios of the line.
4. How are direction cosines and direction ratios related to each other?
Ans. Direction ratios of a line are the ratios of the direction cosines of the line. In other words, the direction ratios are derived from the direction cosines by dividing each direction cosine by their GCF.
5. What is the significance of direction cosines and direction ratios in geometry?
Ans. Direction cosines and direction ratios are important in geometry as they provide a way to describe the orientation and direction of a line in three-dimensional space. They help us understand the relationship between lines and coordinate axes, and are used in various applications such as vector calculations and solving problems involving lines and angles.
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