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Q. 16. Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)            (1998 - 8 Marks)

Solution. The P.Vs. of the points A, B, C, D are

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Equations of AC and BD are

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

For point of intersection say T compare the coefficients

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                ...(1)

Let R and S be mid-points of parallel sides AB and DC then

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Equation of RS by r = a + s(b-a) is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Comparing the coefficients, we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

which is true . Hence proved.


Q. 17. For any two vectors u and v, prove that            (1998 - 8 Marks)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. We have, Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Where θ is the angle between Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a unit vector perpendicular to both  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and is such that Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE form a right handed system.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(b) Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is perpendicular to both u and v , | a |2 = a2

L.H.S. = (1 + a2) (1 + b2)
R.H.S. = (1 - ab cos qθ)2 + (u + v)2 (u × v)2  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

= 1 + a2b2 cos2 θ - 2ab cos θ +a2 +b2 + 2abcos θ + a2b2 sin2 θ.1+ 0

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 18. Let u an d v be unit vector s. If w is a vector such th at w +(w × u) = v, then prove that |(u × v) · w| < 1/2 and  that the equality holds if and only if u is perpendicular to v.            (1999 - 10 Marks)

Solution. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(θ is angle between Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Equality holds when  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 19. Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.              (2001 - 5 Marks)

Solution. Let  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE be the position vectors by A, B, and C respectively,

Let AB, BE and CF be the bisectors of ∠A, ∠B , and ∠C respectively.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

a, b, c are the lengths of sides BC, CA and AB respectively.

Now we know by angle bisector thm that AD divides, BC in the ratio

BD : DC = AB : AC = c : b.

∴ The position vector of D is   Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let I be the point of intersection of BE and AD. Then in ΔABD, BI is bisector of ∠B.

∴ DI : IA = BD : BA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

As p.v. of I is symm. in  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and a,b,c.

∴ It must lie on CF as well.

[We can also see that p.v. of intersection of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Above prove that all the ∠ bisectors pass through I, i.e., these are concurrent.

Q. 20. Find 3-dimensional vectors  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE satisfying Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE              (2001 - 5 Marks)

Ans. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are some possible values.

Solution. Given data is insufficient to uniquely determine the three vectors as there are only 6 equations involving 9 variables.

∴ We can obtain infinitely many set of three vectors,

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  satisfying these conditions. 

From the given data, we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

[where θ is the angle between Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE]

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Now since any two vectors are always coplanar, let us suppose that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are in x-y plane.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is along the positive direction of x-axis  

then  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE    Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE makes an angle 135° with  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE lines in x-y plane, 

Also keeping in mind 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

are some possible answers.


Q. 21. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE where f1, f2, g1, g2 are continuous functions. If  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are nonzero vectors for all t and  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Then show that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  are parallel for some t.           (2001 - 5 Marks)

Solution.Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is parallel to  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE if and only if 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Since h is a continuous function, and h(0).h(1) < 0

⇒ There is some t ∈ [0,1] for which h(t) = 0 i.e.,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are parallel vectors for this t.


Q. 22. Let V be the volume of the parallelopiped formed by the vectors Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE where r = 1, 2, 3, are n on negative real numbers and  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE show that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE             (2002 - 5 Marks)

Solution. 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ V = (a1b2c3 + a2b3c1+ a3b1c2) - (a1b3c2 + a2b1c3+ a3b2c1) ....(1)

Now we know that  AM > GM

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

[same reason]

L3 > V from (1) Hence Proved.


Q. 23. (i) Find the equation of the plane passing through the points (2, 1, 0), (5, 0, 1) and (4, 1, 1).          (2003 - 4 Marks)

(ii) If P is the point (2, 1, 6) then find the point Q such that PQ is perpendicular to the plane in (i) and the mid point of PQ lies on it.

Ans. (i) x + y - 2z = 3
(ii) Q(6, 5, - 2)

Solution. (i) Plane passing thr ough (2, 1, 0) , (5, 0, 1) and (4, 1, 1) is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ ( x - 2)(-1 - 0) - (y - 1)(3- 2) + z(0 - (-2)) = 0
⇒ -x + 2 - y +1+ 2z = 0 ⇒ x + y - 2z = 3

(ii) As per question we have to find a pt. Q such that PQ is ⊥ to the plane x + y - 2z = 3      ...(1)

And mid pt. of PQ lies on the plane, (Clearly we have to find image of pt. P with respect to plane).

Let Q be (α, β,γ)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Eqn of PM passing through P(2, 1, 6) and ⊥ to plane x + y - 2z = 3 , is given by

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

For some value of  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE lies on PM

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
But M lies on plane (1)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 24. If Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are three non-coplanar unit vectors and Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEare the angles between  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEandSubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE respectively andSubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are unit vectors along the bisectors of the angles  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE respectively. Prove that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE               (2003 - 4 Marks)

Solution. Given that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are three non coplanar unit vectors. Angle  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and between  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE represent  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let P be a pt. on angle bisector of ∠AOB such that OAPB is a parallelogram

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Also ∠POA =∠BOP = α /2

∴ ∠APO = ∠BOP = α /2 (alt. int. ∠ 's)

∴ In ΔOAP, OA= AP a unit vector in the direction of  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴A unit vector in the direction of

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
[Using defn of vector triple product.]
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE          ...(1)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE when ever any two vectors are same)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

From (1) and (2),

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 25. If  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEare distinct vectors such that Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                    

Prove that Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                (2004 - 2 Marks)

Solution. Given that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Such that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE        ...(1)
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                    ...(2)
To prove Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subtracting eqn (2) from (1) we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ Angle between Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is either 0 or 180° . 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

as a, b, c, d all are different. Hence Proved.


Q. 26. Find the equation of plane passing through (1, 1, 1) & parallel to the lines L1, L2 having direction ratios (1,0, –1), (1, –1,0). Find the volume of tetrahedron formed by origin and the points where these planes intersect the coordinate axes.           (2004 - 2 Marks)

Ans. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. ∴ The plane is parallel to the lines L1 and L2 with direction ratios as (1, 0, –1) and (1, –1, 0)

∴ A vector perpendicular to L1 and L2 will be parallel to the normal  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE to the plane.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Eqn. of plane through (1, 1, 1) and having normal vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Now the pts where this plane meets the axes are A(3, 0, 0), B(0, 3, 0),C (0, 0, 3)
∴  Vol.of  tetrahedron OABC

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(Note that ΔABC is an equilateral Δ here.)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 27. A parallelopiped ‘S’ h as base points A, B, C an d D and upper face points A', B', C' and D'. This parallelopiped is compressed by upper face A'B'C'D' to form a new parallelopiped ‘T’ having upper face points A'', B'', C'' and D''. Volume of parallelopiped T is 90 percent of the volume of parallelopiped S. Prove that the locus of ‘A''’, is a plane.           (2004 - 2 Marks)

Solution. ATQ ‘S’ is the parallelopiped with base points A, B, C and D and upper face points A', B',C' and D ' . Let its vol. be V.
By compr essi n g it by upper face A', B', C ',D', a n ew parallelopiped ‘T’ is formed whose upper face pts are now A'',B'',C'' and D ''. Let its vol. be VT .

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let h be the height of original parallelopiped S.
Then VS = (ar ABCD)x h     ...(1)

Let equation of plane ABCD be  ax + by +cz +d= and A''(α,β,γ)

Then height of new parallelopipe T is the length of perpendicular from A'' to ABCD

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

But given that,

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE             ...(3)

From (1), (2) and (3) we get,

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

which is a plane parallel to ABCD .  Hence proved.


Q. 28. P1 and P2 are planes passing through origin. L1 and L2 are two line on P1 and P2 respectively such that their intersection is origin. Show that there exists points A, B, C, whose permutation A', B', C' can be chosen such that (i) A is on L1, B on P1 but not on L1 and C not on P1 (ii) A' is on L2, B' on P2 but not on L2 and C' not on P2        (2004 - 4 Marks)

Solution. Following fig. shows the possible situation for planes P1 and P2 and the lines L1 and L2

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Now if we choose pts A, B, C as follows.
A on L1 , B on the line of intersection of P1 and P2 but other than origin and C on L2 again other than origin then we can consider

A corresponds to one of A',B',C' and
B corresponds to one of the remaining of A', B',C' and
C corresponds to third of A',B',C' e.g.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence one permutation of [ABC] is [CBA]. Hence Proved.
Q. 29. Find the equation of the plane containing the line 2x – y + z – 3 = 0, 3x + y + z = 5 and at a distance of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEfrom the point(2, 1, –1)           (2005 - 2 Marks)

Ans. 62 x + 29 y + 19z - 105 = 0

Solution. The given line is 2x - y + z - 3 = 0 = 3x + y + z - 5

Which is intersection line of two planes

2 x - y + z - 3 = 0 ...(i)
and 3x + y + z - 5 = 0 ...(ii)

Any plane containing this line will be the plane passing through the intersection of two planes (i) and (ii).

Thus the plane containing given line can be written as

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

As its distance from the pt.  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Squaring both sides, we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ The required equations of planes are 2x- y+z-3 = 0

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

or 62x + 29 y + 19z - 105 = 0


Q. 30. If the incident ray on a surface is along the unit vectorSubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthe reflected ray is along the unit vector Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  the normal is along unit vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE outwards. Express  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  in terms of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE            (2005 - 4 Marks)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans.  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Given that incident ray is along Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEreflected ray is along Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE normal is along Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEoutwards. The given figure can be redrawn as shown.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

We know  that incident ray, reflected ray and normal lie in a plane, and angle of incidence = angle of reflection.

Therefore Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwill be along the angle bisector of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

[∵ Angle  bisector will along a vector dividing in same ratio as the ratio of the sides forming that angle.]

But Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a unit vector.

Where Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Substituting this value in equation (1) we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The document Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 2 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Maths 35 Years JEE Main & Advanced Past year Papers.
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