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Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

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Q. 1. From a point O inside a triangle ABC, perpendiculars OD, OE, OF are drawn to the sides BC, CA, AB respectively. Prove that the perpendiculars from A, B, C to the sides EF, FD, DE are concurrent.            (1978)

Solution. Let with respect to O, position vectors of points A, B, C, D, E, F be  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Let perpendiculars from A to EF  and from B to DF meet each other at H. Let position vector of H be  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE we join CH.
In order to prove the statement given in question, it is sufficient to prove that CH is perpendicular to DE.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Adding (4) and (5), we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

(using (1), (2) and (3))

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Q. 2. A1, A2,...................... An are the vertices of a regular plane polygon with n sides and O is its centre. Show that Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE         (1982 - 2 Marks)

Solution.  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE all vectors are of same magnitude, say ‘a’  and angle between any two consecutive vector  is same  that is Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE be the unit vectors ⊥ to the plane of the polygon.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE


Q. 3. Find all values of λ such that x, y,z, ≠ (0, 0, 0) and Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEESubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE  are unit vectors along the coordinate axes.             (1982 - 3 Marks)

Ans. λ = 0, - 1

Solution.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
All the above three equations are satisfied for x, y, z not all zero if

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE


Q. 4. A vector Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE has components A1, A2, A3 in a right -handed rectangular Cartesian coordinate system oxyz. The coordinate system is rotated about the x-axis through an angle Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE Find the components of A in the new coordinate system, in terms of A1, A2, A3.

Ans. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Solution. Since vector Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE has components A1 , A2 ,A3, in the coordinate system OXYZ,

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

When given system is rotated through Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE  the new x-axis is along old y-axis and new y-axis is along the old negative x-axis z remains same as before.
Hence the components of A in the new system are

A2 , - A1,A3 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE


Q. 5. The position vectors of the points A, B, C an d D are Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEErespectively. If the points A, B, C and D lie on a plane, find the value of λ .

Ans. 146/17

Solution.Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

We know that A, B, C, D lie in a plane if  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE are coplanar i.e.  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

 
Q. 6. If A, B, C , D are any four points in space, prove that –

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE(area of triangle ABC)

Solution. Let the position vectors of points A, B, C, D be a, b, c, and d respectively with respect to some origin O.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Then,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEESubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE             …(1)
Also Area of ΔABC is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE       … (2)

From (1) and (2), we ge

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Q. 7. Let OA CB be a parallelogram with O at the origin and OC a diagonal. Let D be the midpoint of OA. Using vector methods prove that BD and CO intersect in the same ratio. Determine this ratio.

Solution. OACB is a parallelogram with O as  origin. Let with respect to O position vectors of A and B be  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE respectively..
Then p.v. of C is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Also D is mid pt. of OA, therefore position vector of D is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

CO and BD intersect each other at P.

Let P divides CO in the ratio λ : 1 and BD in the ratio μ : 1 Then by section theorem, position vector of pt. P dividing CO in ratio

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE             …(1)
And position vector of pt. P dividing BD in the ratio μ : 1 is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE              …(2)

As (1) and (2) represent the position vector of same point, we should have

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Equating the coefficients of  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE  we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE           … (i)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE               …(ii)

From (ii) we get λ = μ ⇒ P divides CO and BD in the same ratio.

Putting λ = μ in eq. (i) we get μ = 2

Thus required ratio is 2 : 1.


Q. 8. If vectors Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE are coplanar, show that

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Solution. Given that Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE are  three coplanar vectors.

∴ There exists scalars x, y, z, not all zero, such that

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE               … (1)

Taking dot product of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE we get 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE             … (2)

Again taking dot product of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE            … (3)

Now equations (1), (2), (3) form a homogeneous system of equations, where x, y, z are not all zero.

∴ system must have non trivial solution and for this, determinant of coefficient matrix should be zero

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE                Hence Proved.

Q. 9. In a triangle OAB, E is the midpoint of BO and D is a point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, determine the ratio OP : PD using vector methods.

Solution. With O as origin let  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE be the position vectors of A and B respectively.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Then the position vector of E, the mid point of OB is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Again since AD : DB = 2 : 1, the  position vector of D is 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

∴ Equation of OD is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE                …(1)

and  Equation of AE is 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE            …(2)

If OD and AE intersect at P, then we will have identical values of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE Hence comparing the coefficients of  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE we get 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Putting value of t in eq. (1) we get position vector of point of intersection P as

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE                    … (3)

Now if P divides OD in the ratio λ : 1, then p.v. of P is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE                … (4)

From (3) and (4) we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE


Q. 10. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEESubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE Determine a vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE Satisfying  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Ans. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Solution. We are given that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE and  to determine a vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE such that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE and  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

⇒ y -z =-10 … (1)
z -x =-11 … (2)
x -y= 7 … (3)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

⇒ 2x +z=0 … (4)

Substituting y =x- 7 and z = -2x from (3) and (4) respectively in eq. (1) we get

x - 7 + 2x = -10 ⇒ 3x =-3

⇒ x =-1 , y =-8 and z = 2

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE


Q. 11. Determine the value of ‘c’ so that for all real x, the vector Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE  make an obtuse angle with each other.

Ans. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Solution. We have,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Now we know that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

As angle between  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE is obtuse, therefore 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE


Q. 12. In a triangle ABC, D an d E are points on BC and AC respectively, such that BD = 2 DC and AE = 3EC. Let P be the point of intersection of AD and BE. Find BP/PE using vector methods.

Ans. 8 : 3

Solution. Let  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE be the position vectors of pt A, B and C respectively with respect to some origin.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

ATQ, D divides BC in the ratio 2 : 1 and E divides AC in the ratio 3 : 1.

∴ position vector of D is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE and position vector of E is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Let pt. of intersection P of AD and BE divides BE in the ratio k : 1 and AD in the ratio m : 1, then position vectors of P in these two cases are
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Equating the position vectors of P in two cases we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE             … (1)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE                   … (2)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Dividing (3) by (2) we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE   the req. ratio is 8 : 3.


Q. 13. If the vectors  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE are not coplanar, then prove that the vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEESubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE parallel to  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Solution. Given that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE are  not coplanar Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Consider,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEESubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Here,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE                 …(2)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE                …(3)

[NOTE :  Here we have tried  to write the given expression in such a way that we can get terms involving Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE other terms similar which can get cancelled.]

Adding (1), (2) and (3), we get given vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

⇒ given vector = some constant multiple of  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

⇒ given vector is parallel to  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE


Q. 14. The position vectors of the vertices A, B an d C of a tetrahedron ABCD are Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE respectively. The altitude from vertex D to the opposite face ABC meets the median line through A of the triangle ABC at a point E. If the length of the side AD is 4 and the volume of the tetrahedron is Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE find the position vector of the point E for all its possible positions.

Ans. (-1, 3, 3) or (3, - 1, - 1)

Solution. We are given AD = 4

Volume of tetrahedron  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

We have to find the P.V. of point E.  Let it divides median AF in the ratio λ : 1

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE            …(3)
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE


Q. 15. If A, B and C are vectors such that | B | = | C |. Prove that [(A + B) × (A + C)] × (B × C) (B + C)  = 0 .

Solution. We have,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEESubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE
[∵ (a x b) x c = (a.c)b- (b.c)a]

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

[∵ [A B C] = 0 if any two of A, B, C are equal.]

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Thus, LHS of the given expression

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE          [∵| B | = C|]

The document Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced Notes | Study Mathematics For JEE - JEE is a part of the JEE Course Mathematics For JEE.
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