Show that the four points A , B , C , D with position vectors a , → b , → c , → d → respectively such that 3 a → - 2 b → + 5 c → - 6 d → = 0 , are coplanar. Also, find the position vector of the point of intersection of the line segments A C and B D .

Let A C and B D intersects at a point P . We have, 3 a → - 2 b → + 5 c → - 6 d → = 0 → . ⇒ 3 a → + 5 c → = 2 b → + 6 d → Since sum of coefficients on both sides of the above equation is 8 , so we divide the equation on both sides by 8 . i.e., 3 a → + 5 c → 8 = 2 b → + 6 d → 8 ⇒ 3 a → + 5 c → 3 + 5 = 2 b → + 6 d → 2 + 6 Therefore, P divides A C in the ratio of 3 : 5 and P divides B D in the ratio of 2 : 6 Therefore, position vector of the point of intersection of A C and B D will be 3 a → + 5 c → 8 or 2 b → + 6 d → 8 . Since A C and B D intersect at point P , therefore A , B , C and D are coplanar.

Show that the four points A , B , C , D with position vectors a , → b , → c , → d → respectively such that 3 a → - 2 b → + 5 c → - 6 d → = 0 , are coplanar. Also, find the position vector of the point of intersection of the line segments A C and B D .

Let A C and B D intersects at a point P . We have, 3 a → - 2 b → + 5 c → - 6 d → = 0 → . ⇒ 3 a → + 5 c → = 2 b → + 6 d → Since sum of coefficients on both sides of the above equation is 8 , so we divide the equation on both sides by 8 . i.e., 3 a → + 5 c → 8 = 2 b → + 6 d → 8 ⇒ 3 a → + 5 c → 3 + 5 = 2 b → + 6 d → 2 + 6 Therefore, P divides A C in the ratio of 3 : 5 and P divides B D in the ratio of 2 : 6 Therefore, position vector of the point of intersection of A C and B D will be 3 a → + 5 c → 8 or 2 b → + 6 d → 8 . Since A C and B D intersect at point P , therefore A , B , C and D are coplanar.

Show that the four points P , Q , R , S with position vectors p → , q → , r → , s → respectively such that 5 p → - 2 q → + 6 r → - 9 s → = 0 → ,are coplanar. Also, find the position vector of the point of intersection of the line segments P R and Q S .

Let the point of intersection of the line segments P R and Q S is A . Then 5 p → - 2 q → + 6 r → - 9 s → = 0 → ⇒ 5 p → + 6 r → = 2 q → + 9 s → The sum of the coefficients on both the sides of the above equation is 11 .So, we divide the given equation with 11 . ⇒ 5 p → + 6 r → 11 = 2 q → + 9 s → 11 ⇒ 5 p → + 6 r → 5 + 6 = 2 q → + 9 s → 2 + 9 Therefore, A divides P R in the ratio of 5 : 6 and Q S in the ratio of 2 : 9 The position vector of the point of intersection of the line segment is 5 p + 6 r ' 11 , 2 q + 9 s 11 .

E M B I B E

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 4

HARD

12th CBSE

IMPORTANT

Earn 100

Show that the four points $A, B, C, D$ with position vectors $\vec{a,} \vec{b,} \vec{c,} \vec{d}$ respectively such that $3 \vec{a} - 2 \vec{b} + 5 \vec{c} - 6 \vec{d} = 0,$ are coplanar. Also, find the position vector of the point of intersection of the line segments $A C$ and $B D .$

Important Questions on Algebra of Vectors

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 5

Show that the four points

P, Q, R, S

with position vectors

\vec{p}, \vec{q}, \vec{r}, \vec{s}

respectively such that

5 \vec{p} - 2 \vec{q} + 6 \vec{r} - 9 \vec{s} = \vec{0}

,are coplanar. Also, find the position vector of the point of intersection of the line segments

P R

and

Q S

.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 6

The vertices

A, B, C

of triangle

A B C

have respectively position vectors

\vec{a}, \vec{b}, \vec{c}

with respect to a given origin

O

. Show that the point

D

where the bisector of

∠ A

meets

B C

has position vector

\vec{d} = \frac{β \vec{b} + γ \vec{c}}{β + γ}, where β = |\vec{c} - \vec{a}| and, γ = |\vec{a} - \vec{b}|

.Hence, deduce that the incentre

I

has position vector

\frac{α \vec{a} + β \vec{b} + γ \vec{c}}{α + β + γ}, where α = |\vec{b} - \vec{c}|

.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 1

If

O

is a point in space

A B C

is a triangle and

D, E, F

are the mid-points of the sides

B C, C A

and

A B

respectively of the triangle, prove that

\vec{O A} + \vec{O B} + \vec{O C} = \vec{O D} + \vec{O E} + \vec{O F}

.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 2

Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 3

A B C D

is a parallelogram and

P

is the point of intersection of its diagonals. If

O

is the origin of reference, show that

\vec{O A} + \vec{O B} + \vec{O C} + \vec{O D} = 4 \vec{O P}

.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 4

Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 5

$A B C D$ are four points in a plane and $Q$ is the point of intersection of the lines joining the mid-points of $A B$ and $C D; B C$ and $A D$ . Show that $\vec{P A} + \vec{P B} + \vec{P C} + \vec{P D} = 4 \vec{P Q}$ where $P$ is any point.