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Solved Subjective Question on Vector Set 3

Posted on - 04-01-2017

Math

IIT JEE

Example.1

Two system of forces P, Q, R and P¢, Q¢, R¢ act along the side BC, CA, AB of a DABC. Prove that the resultant will be parallel if = 0.

Solution

Unit vector along

=cos (p – C)+sin(p– C)

= – cosC+ sinC

Unit vector along

=cos(p + B)+ sin(p + B)

= – cos B – sinB

If S and S¢ be resultant in the two cases, then

= P+ Q (– cos C+ sin C ) + R( – cosB– sinB) ….(1)

Similarly

S¢ = (P¢ – Q¢ cosC – R¢ cosB) + (Q¢ sinC – R¢ sinB) ….(2)

If Q and Q¢ be the angles made by the resultant with x–axis, then

tanq = and tanq¢ =

if the resultant are parallel , then q = q¢ \ tanq = tanq¢

upon solving we get


&⇒
(PQ¢ – P¢Q) sinC + (RP¢ – R¢P) sinB + (QR¢ – Q¢R) sinA = 0

= 0.

Example.2

Line L1 is parallel to a vector and passes through a point A (7,6,2) and the line L2 is parallel to a vector and passes through a point B (5, 3,4). Now a line L3 parallel to a vector intersects the lines L1 and L2 at points C and D respectively. Find .

Solution

P.V. of C.

a∈R

P.V. of D, b ∈ R

and we know that

Hence by comparing both we get3a + 2b – 2c = 2-€‘2a + b+ 2c = 3


&⇒
-€‘4a + 3b + c = -€‘ 2
&⇒
a = 2, b = 1, c = 3


&⇒

Example.3

A circle is inscribed in an n-sided regular polygon a1 , a2., . . . ., an having each side a unit. For any arbitrary point p on the circle, prove that .

Solution

Let the centre of the incircle be the reference point. Then

[

= nR2 +nr2 – 2

=

now R =

R2 + r2 = =


&⇒
.

Example.4

In a triangle PQR, S and T are points on QR and PR respectively, such that QS =3SR and PT = 4TR. Let M be the point of intersection of PS and QT. Determine the ratio QM:MT using vector methods. .

Solution

Let QM : MT = l : 1 and PM : MS = m :1 and .

Now


&⇒
…(1)

Also ….(2)

From (1) and (2), =

On comparing, we get …..(3)

and …..(4)


&⇒

&⇒
m = 16/3 and l = 15/4

So QM :MT = 15 :4. .

Example.5

Let be non-coplanar unit vectors, equally inclined to one another at an angle q. If , find the scalars p, q and r in terms of q.

Solution

Taking dot product with , we get

p cosq + r cosq + q = 0 .....(1).

p cosq + q cosq + r = ..... (2)

p + q cosq + r cosq = ..... (3)

1 - cosq = p ......(4)

We get (p, q, r) =

Or, (p, q, r) = .

 
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