Find the magnitude and direction of r → 1 - r → 2 when r → 1 = r → 2 = 5 and r 1 → points North –East while points r 2 → North –West.

To find the required magnitude and direction of r 1 → - r 2 → . Here, r 1 → = O A → pointing North-East and r 2 → = O B → pointing North-West. Where O A → = O B → = 5 . As shown in figure, ∠ B O A = 90 ° r 1 → - r 2 → = B A → (Using triangle law) Clearly, ∠ B O A is right-angled at O . B A → 2 = O A → 2 + O B → 2 =5 2 +5 2 =50 B A → = 5 5 Or r 1 → - r 2 → = 5 5 i.e., r 1 → – r 2 → has magnitude 5 5 and points from West to East.

Let O A C B be a parallelogram with O → at the origin and O C → as a diagonal. Let D → be the midpoint of O A → . Using vector methods, prove that B D → and C O → intersect in the same ratio, determine this ratio.

To find out the ratio, it is given that O A C B be a parallelogram shown as Here, O D → = 1 2 B C → ⇒ O P → + P D → = 1 2 B P → + P C → ; [using triangle law of vector addition] ⇒ 2 O P → + 2 P D → = B P → + P C → ⇒ – 2 P O → + 2 P D → = – P B → + P C → ⇒ P B → + 2 P D → = P C → + 2 P O → ⇒ P B → +2 P D → 1 +2 = P C → +2 P O → 1 +2 The common point P → of B D → and C O → divides each in the ratio 2 : 1 .

∆ A B C is a triangle with the point P on side B C → such that 3 B P → = 2 P C → , the point Q is on the line C A → such that 4 C Q → = Q A → . Find the ratio in which the line joining the common point R of A P → and B Q → and the point S divides A B → .

It is given that A B C is a triangle with the point P on side B C → such that 3 B P → = 2 P C → , the point Q is on the line C A → such that 4 C Q → = Q A → . Let S be point of intersection of A B → and C R → . Let A be the origin 0 → and the position vectors of the points B , C , P , Q , R and S be b → , c → , p → , q → , r → and s → , respectively. ⇒ p → = 3 b → +2 c → 5 . . . i And q → = 4 c → 5 . . . ii Equating c → from equation i & ii ⇒ 5 p → -3 b → 2 = 5 q → 4 ⇒ 10 p → – 6 b → =5 q → 10 p → = 5 q → + 6 b → ⇒ 10 p → 11 = 5 q → +6 b → 11 = r → . . . iii Equating p → from equation i & iii ⇒ 11 r → 10 = p → = 3 b → +2 c → 5 11 r → = 6 b → + 4 c → ⇒ 11 r → - 4 c → = 6 b → 11 r → –4 c → 7 = 6 7 b → = s → , Thus S divides A B → in ratio 6 : 1

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IMPORTANT

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Let $A B C$ be a triangle, the position vectors whose vertices are $7 \hat{j} + 10 \hat{k}, - \hat{i} + 6 \hat{j} + 6 \hat{k}$ and $- 4 \hat{i} + 9 \hat{j} + 6 \hat{k}$ . Then, $Δ A B C$ is

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Important Questions on Vector Algebra

EASY

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Vector Algebra Exercise 2>Q 86

The sides of a parallelogram are

2 \hat{i} + 4 \hat{j} - 5 \hat{k}

and

\hat{i} + 2 \hat{j} + 3 \hat{k}

. The unit vector parallel to one of the diagonals is

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Vector Algebra Exercise 2>Q 87

A (- 4, 0, 3)

and

B (14, 2, - 5)

, then which one of the following points lies on the bisector of the angle between

\vec{O A}

and

\vec{O B}

(

O

is the origin of reference)?

EASY

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Vector Algebra Exercise 2>Q 88

If point

\hat{i} + \hat{j}, \hat{i} - \hat{j}

and

p \hat{i} + q \hat{j} + r \hat{k}

are collinear, then

EASY

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Vector Algebra Exercise 2>Q 89

\vec{a}, \vec{b}

and

\vec{c}

are non-coplanar vectors and

λ

is a real number, then the vectors

(\vec{a} + 2 \vec{b} + 3 \vec{c}), (λ \vec{b} + μ \vec{c})

and

(2 λ - 1) \vec{c}

are coplanar when

EASY

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Vector Algebra Exercise 7>Q 112

A vector

\vec{a}

has components

{\vec{a}}_{1}, {\vec{a}}_{2} & {\vec{a}}_{3}

in a right-handed rectangular Cartesian system

O X Y Z

. The coordinate system is rotated about

Z

-axis through the angle

\frac{π}{2}

. Find the components of

\vec{a}

in new system.

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Vector Algebra Exercise 7>Q 113

Find the magnitude and direction of

{\vec{r}}_{1} - {\vec{r}}_{2}

when

|{\vec{r}}_{1}| = |{\vec{r}}_{2}| = 5

and

\vec{r_{1}}

points North –East while points

\vec{r_{2}}

North –West.

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Vector Algebra Exercise 7>Q 114

Let

O A C B

be a parallelogram with

\vec{O}

at the origin and

\vec{O C}

as a diagonal. Let

\vec{D}

be the midpoint of

\vec{O A}

. Using vector methods, prove that

\vec{B D}

and

\vec{C O}

intersect in the same ratio, determine this ratio.

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Vector Algebra Exercise 7>Q 115

∆ A B C

is a triangle with the point

P

on side

\vec{B C}

such that

3 \vec{B P} = 2 \vec{P C}

, the point

Q

is on the line

\vec{C A}

such that

4 \vec{C Q} = \vec{Q A}

. Find the ratio in which the line joining the common point

R

\vec{A P}

and

\vec{B Q}

and the point

S

divides

\vec{A B}

Let ABC be a triangle, the position vectors whose vertices are 7j^+10k^,-i^+6j^+6k^ and -4i^+9j^+6k^. Then, ΔABC is

Important Questions on Vector Algebra

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Let $A B C$ be a triangle, the position vectors whose vertices are $7 \hat{j} + 10 \hat{k}, - \hat{i} + 6 \hat{j} + 6 \hat{k}$ and $- 4 \hat{i} + 9 \hat{j} + 6 \hat{k}$ . Then, $Δ A B C$ is