Consider the points A ( 1 , 2 , 7 ) , B ( 2 , 6 , 3 ) , C ( 3 , 10 , - 1 ) . Find A B → , B C → .

The given points are A ( 1 , 2 , 7 ) , B ( 2 , 6 , 3 ) , C ( 3 , 10 , - 1 ) . Position vector of A is O A → = i ^ + 2 j ^ + 7 k ^ Position vector of B is O B → = 2 i ^ + 6 j ^ + 3 k ^ Position vector of C is O C → = 3 i ^ + 10 j ^ - k ^ Now, A B → = O B → - O A → = 2 i ^ + 6 j ^ + 3 k ^ - i ^ + 2 j ^ + 7 k ^ ∴ A B → = i ^ + 4 j ^ - 4 k ^ and B C → = O C → - O B → = 3 i ^ + 10 j ^ - k ^ - 2 i ^ + 6 j ^ + 3 k ^ ∴ B C → = i ^ + 4 j ^ - 4 k ^

The position vector of three points A , B , C are given to be i ^ + 3 j ^ + 3 k ^ , 4 i ^ + 4 k ^ and - 2 i ^ + 4 j ^ + 2 k ^ respectively. Find A B → and A C → .

Given the position vectors of three points A , B , C as i ^ + 3 j ^ + 3 k ^ , 4 i ^ + 4 k ^ and - 2 i ^ + 4 j ^ + 2 k ^ . Now, A B → = Position vector of B - Position vector of A . ⇒ A B → = 4 i ^ + 4 k ^ - i ^ + 3 j ^ + 3 k ^ ⇒ A B → = 3 i ^ - 3 j ^ + k ^ and A C → = Position vector of C - Position vector of A ⇒ A C → = - 2 i ^ + 4 j ^ + 2 k ^ - i ^ + 3 j ^ + 3 k ^ ⇒ A C → = - 3 i ^ + j ^ - k ^

Show that the four points with position vectors 4 i ^ + 8 j ^ + 12 k ^ , 2 i ^ + 4 j ^ + 6 k ^ , 3 i ^ + 5 j ^ + 4 k ^ and 5 i ^ + 8 j ^ + 5 k ^ are coplanar.

Let A → = 4 i ^ + 8 j ^ + 12 k ^ , B → = 2 i ^ + 4 j ^ + 6 k ^ , C → = 3 i ^ + 5 j ^ + 4 k ^ & D → = 5 i ^ + 8 j ^ + 5 k ^ Now, A B → = Position vector of B - Position vector of A = 2 i ^ + 4 j ^ + 6 k ^ - 4 i ^ + 8 j ^ + 12 k ^ = - 2 i ^ - 4 j ^ - 6 k ^ Similarly, A C → = 3 i ^ + 5 j ^ + 4 k ^ - 4 i ^ + 8 j ^ + 12 k ^ = - i ^ - 3 j ^ - 8 k ^ and A D → = 5 i ^ + 8 j ^ + 5 k ^ - 4 i ^ + 8 j ^ + 12 k ^ = i ^ - 7 k ^ If A , B , C , D are coplanar, A B → , A C → , A D → lie in the same plane and are coplanar and A B → A C → A D → = 0 Now, A B → A C → A D → = - 2 - 4 - 6 - 1 - 3 - 8 1 0 - 7 = - 2 21 - 0 + 4 7 - - 8 - 6 0 - - 3 = - 42 + 60 - 18 = 0 Hence, A , B , C , D are coplanar.

Show that the points A , B , C , D with position vectors 4 i ^ + 8 j ^ + 12 k ^ , 2 i ^ + 4 j ^ + 6 k ^ , 3 i ^ + 5 j ^ + 4 k ^ and 5 i ^ + 8 j ^ + 5 k ^ respectively are coplanar.

Let A → = 4 i ^ + 8 j ^ + 12 k ^ , B → = 2 i ^ + 4 j ^ + 6 k ^ , C → = 3 i ^ + 5 j ^ + 4 k ^ & D → = 5 i ^ + 8 j ^ + 5 k ^ Now, A B → = Position vector of B - Position vector of A = 2 i ^ + 4 j ^ + 6 k ^ - 4 i ^ + 8 j ^ + 12 k ^ = - 2 i ^ - 4 j ^ - 6 k ^ Similarly, A C → = 3 i ^ + 5 j ^ + 4 k ^ - 4 i ^ + 8 j ^ + 12 k ^ = - i ^ - 3 j ^ - 8 k ^ and A D → = 5 i ^ + 8 j ^ + 5 k ^ - 4 i ^ + 8 j ^ + 12 k ^ = i ^ - 7 k ^ If A , B , C , D are coplanar, A B → , A C → , A D → lie in the same plane and are coplanar and A B → A C → A D → = 0 Now, A B → A C → A D → = - 2 - 4 - 6 - 1 - 3 - 8 1 0 - 7 = - 2 21 - 0 + 4 7 - - 8 - 6 0 - - 3 = - 42 + 60 - 18 = 0 Hence, A , B , C , D are coplanar.

EASY

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What is antiparallel vectors?

Important Questions on Motion in a Plane

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Consider the points $A (1, 2, 7), B (2, 6, 3), C (3, 10, - 1)$ .

Find $\vec{A B}, \vec{B C}$ .

HARD

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Show that the position vector of the point

C

, which divides the line joining the points

A and B

having position vectors

\vec{a} and \vec{b}

internally in the ratio

m : n is \frac{m \vec{b} + n \vec{a}}{m + n}

HARD

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Find

x

such that the four points

A (3, 2, 1), B (4, x, 5), C (4, 2, - 2) and D (6, 5, - 1)

are coplanar.

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The position vector of three points

A, B, C

are given to be

\hat{i} + 3 \hat{j} + 3 \hat{k}, 4 \hat{i} + 4 \hat{k}

and

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

respectively. Find

\vec{A B}

and

\vec{A C}

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

A person goes

2 km

east, then

3 km

north, then

4 km

west and then

1 km

north, starting from the origin. This point is taken as vector

A

The vector

B

B such that

3 A + 5 B = (9, 32),

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The position vector of the point which divides the join of the points

2 \vec{a} - 3 \vec{b}

and

\vec{a} + \vec{b}

in the ratio of

3 : 1

internally is

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Show that the four points with position vectors

4 \hat{i} + 8 \hat{j} + 12 \hat{k}

2 \hat{i} + 4 \hat{j} + 6 \hat{k}, 3 \hat{i} + 5 \hat{j} + 4 \hat{k}

and

5 \hat{i} + 8 \hat{j} + 5 \hat{k}

are coplanar.

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Force

\vec{F}

applied on a body is written as

\vec{F} = (\hat{n} \times \vec{F}) \cdot \hat{n} + \vec{G}

, where

\hat{n}

is a unit vector. The vector

\vec{G}

is equal to

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Find the component of vector

\vec{P} = 2 \hat{i} + 3 \hat{j}

along the direction of vector

\vec{Q} = \hat{i} + \hat{j} .

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Find the vector joining the points

P (2, 3, 0)

and

Q (- 1, - 2, - 4)

directed from

P

Q

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Two vectors

\vec{A}

and

\vec{B}

have equal magnitudes. The magnitude of

(\vec{A} + \vec{B})

n

times the magnitude of

(\vec{A} - \vec{B})

. The angle between

\vec{A}

and

\vec{B}

is:

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Let

\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} + 2 \hat{j} + \hat{k}

and

\vec{c} = 5 \hat{i} + \hat{j} - \hat{k}

be three vectors. The area of the region formed by the set of points whose position vectors

\vec{r}

satisfy the equations

\vec{r} \cdot \vec{a} = 5

and

| \vec{r} - \vec{b} | + | \vec{r} - \vec{c} | = 4

is closest to the integer.

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Two forces

P

and

Q

, of magnitude

2 F

and

3 F

, respectively, are at an angle

θ

with each other. If the force

Q

is doubled, then their resultant also gets doubled. Then, the angle

θ

is:

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Show that the points

A, B, C, D

with position vectors

4 \hat{i} + 8 \hat{j} + 12 \hat{k}

2 \hat{i} + 4 \hat{j} + 6 \hat{k}, 3 \hat{i} + 5 \hat{j} + 4 \hat{k}

and

5 \hat{i} + 8 \hat{j} + 5 \hat{k}

respectively are coplanar.

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

A vector

\vec{A}

is rotated by a small angle

∆ θ

radians

(∆ θ ≪ 1)

to get a new vector

\vec{B}

. In that case

|\vec{B} - \vec{A}|

is :

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Vector

\vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k}

and

\vec{b} = \hat{i} - \hat{j} + \hat{k}

. What is the unit vector along

\vec{a} + \vec{b}

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The angle, between

\vec{A}

and the resultant of

2 \vec{A} + 3 \vec{B}

and

4 \vec{A} - 3 \vec{B}

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Let

\hat{i}

and

\hat{j}

be the unit vectors along

x

and

y

directions. Then the magnitude of

\hat{i} + \hat{j}

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The unit vector perpendicular to the plane of

\vec{A} = \hat{i} - 3 \hat{j} - \hat{k}

and

\vec{B} = 2 \hat{i} + \hat{j} - \vec{k}

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The unit vector

(a \hat{i} + b \hat{j})

is perpendicular to

(\hat{i} + \hat{j}) .

The value of

b

can be:

What is antiparallel vectors?

Important Questions on Motion in a Plane

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