If a ¯ , b ¯ , c ¯ are non-coplanar vectors, then show that the points with the position vectors 3 a ¯ + 4 b ¯ - 2 c ¯ , a ¯ - 2 b ¯ + 3 c ¯ , a ¯ - 6 b ¯ + 6 c ¯ and 2 a ¯ + 3 b ¯ - c ¯ are coplanar.

Let p , q , r , s be the position vectors of the points P , Q , R and S respectively. p = 3 a ¯ + 4 b ¯ - 2 c ¯ q = a ¯ - 2 b ¯ + 3 c ¯ r = a ¯ - 6 b ¯ + 6 c ¯ s = 2 a ¯ + 3 b ¯ - c ¯ Therefore, P Q = q - p = a ¯ - 2 b ¯ + 3 c ¯ - 3 a ¯ - 4 b ¯ + 2 c ¯ = - 2 a - 6 b + 5 c P R = r - p = a ¯ - 6 b ¯ + 6 c ¯ - 3 a ¯ - 4 b ¯ + 2 c ¯ = - 2 a - 10 b + 8 c P S = s - p = 2 a ¯ + 3 b ¯ - c ¯ - 3 a ¯ - 4 b ¯ + 2 c ¯ = - a - b + c Now, P R × P S = - 2 a - 10 b + 8 c × - a - b + c = - 2 a × - a - b + c - 10 b × - a - b + c + 8 c × - a - b + c = 2 a × a - 2 a × - b - 2 a × c + 10 b × a + 10 b × b - 10 b × c + 8 c × - a + 8 c × - b + 8 c × c Using the property, a × a = 0 , a × b = - b × a P R × P S = 2 a × b - 2 a × c - a × 10 b - 10 b × c + a × 8 c + b × 8 c = - 8 a × b + 6 a × c - 2 b × c ∴ P Q . P R × P S = - 2 a - 6 b + 5 c × - 8 a × b + 6 a × c - 2 b × c = - 2 a . - 8 a × b + 6 a × c - 2 b × c - 6 b . - 8 a × b + 6 a × c - 2 b × c + 5 c . - 8 a × b + 6 a × c - 2 b × c = 16 a . a × b - 12 a . a × c + 4 a . b × c + 48 b . a × b - 36 b . a × c + 12 b . b × c - 35 c . a × b + 30 c . a × c - 10 c . b × c = 16 a a b - 12 a a c + 4 a b c + 48 b a b - 36 b a c + 12 b b c - 40 c a b + 30 c a c - 10 c b c = 16 0 - 12 0 + 4 a b c + 48 0 + 36 a b c + 12 0 - 40 a b c + 30 0 - 10 0 = 0 ∴ P Q , P R , P S are coplanar. ∴ The points P , Q , R and S are coplanar.

If a ¯ , b ¯ , c ¯ are non-coplanar vectors, then show that the points with the position vectors 3 a ¯ + 4 b ¯ - 2 c ¯ , a ¯ - 2 b ¯ + 3 c ¯ , a ¯ - 6 b ¯ + 6 c ¯ and 2 a ¯ + 3 b ¯ - c ¯ are coplanar.

Let p , q , r , s be the position vectors of the points P , Q , R and S respectively. p = 3 a ¯ + 4 b ¯ - 2 c ¯ q = a ¯ - 2 b ¯ + 3 c ¯ r = a ¯ - 6 b ¯ + 6 c ¯ s = 2 a ¯ + 3 b ¯ - c ¯ Therefore, P Q = q - p = a ¯ - 2 b ¯ + 3 c ¯ - 3 a ¯ - 4 b ¯ + 2 c ¯ = - 2 a - 6 b + 5 c P R = r - p = a ¯ - 6 b ¯ + 6 c ¯ - 3 a ¯ - 4 b ¯ + 2 c ¯ = - 2 a - 10 b + 8 c P S = s - p = 2 a ¯ + 3 b ¯ - c ¯ - 3 a ¯ - 4 b ¯ + 2 c ¯ = - a - b + c Now, P R × P S = - 2 a - 10 b + 8 c × - a - b + c = - 2 a × - a - b + c - 10 b × - a - b + c + 8 c × - a - b + c = 2 a × a - 2 a × - b - 2 a × c + 10 b × a + 10 b × b - 10 b × c + 8 c × - a + 8 c × - b + 8 c × c Using the property, a × a = 0 , a × b = - b × a P R × P S = 2 a × b - 2 a × c - a × 10 b - 10 b × c + a × 8 c + b × 8 c = - 8 a × b + 6 a × c - 2 b × c ∴ P Q . P R × P S = - 2 a - 6 b + 5 c × - 8 a × b + 6 a × c - 2 b × c = - 2 a . - 8 a × b + 6 a × c - 2 b × c - 6 b . - 8 a × b + 6 a × c - 2 b × c + 5 c . - 8 a × b + 6 a × c - 2 b × c = 16 a . a × b - 12 a . a × c + 4 a . b × c + 48 b . a × b - 36 b . a × c + 12 b . b × c - 35 c . a × b + 30 c . a × c - 10 c . b × c = 16 a a b - 12 a a c + 4 a b c + 48 b a b - 36 b a c + 12 b b c - 40 c a b + 30 c a c - 10 c b c = 16 0 - 12 0 + 4 a b c + 48 0 + 36 a b c + 12 0 - 40 a b c + 30 0 - 10 0 = 0 ∴ P Q , P R , P S are coplanar. ∴ The points P , Q , R and S are coplanar.

Let α ∈ R and the three vectors a → = α i ^ + j ^ + 3 k ^ , b → = 2 i ^ + j ^ - α k ^ and c → = α i ^ - 2 j ^ + 3 k ^ . Then the set S = { α : a → , b → and c → are coplanar}

contains exactly two positive numbers

contains exactly two numbers only one of which is positive

If a → , b → , c → are non-coplaner vectors such that b → × c → = a → ; c → × a → = b → ; a → × b → = c → , then which of the following is not TRUE?

| a → | = | b → | = | c → | = 2

| a → | | b → | | c → | = 1

HARD

Earn 100

If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors, then show that the points with the position vectors $3 \bar{a} + 4 \bar{b} - 2 \bar{c}, \bar{a} - 2 \bar{b} + 3 \bar{c}, \bar{a} - 6 \bar{b} + 6 \bar{c}$ and $2 \bar{a} + 3 \bar{b} - \bar{c}$ are coplanar.

Important Questions on Addition of Vectors

MEDIUM

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

If the vectors

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

and

\vec{γ} = \hat{i} + c \hat{j} + c^{2} \hat{k}

are three non-coplanar vectors and

|\begin{array}{l} a & a^{2} & 1 + a^{3} \\ b & b^{2} & 1 + b^{3} \\ c & c^{2} & 1 + c^{3} \end{array}| = 0,

then the value of

a b c

HARD

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

Let

S

be the set of all real values of

λ

such that a plane passing through the points

(- λ^{2}, 1, 1), (1, - λ^{2}, 1)

and

(1, 1, - λ^{2})

also passes through the point

(- 1, - 1, 1)

. Then

S

is equal to :

HARD

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

The position vectors of the points

A, B, C

and

D

are

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

and

4 \hat{i} - \hat{j} + λ \hat{k},

respectively. If the points

A, B, C

and

D

lie on a plane, the value of

λ

EASY

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

\vec{a} = \hat{i} + λ \hat{j} + 2 \hat{k} & \vec{b} = μ \hat{i} + \hat{j} - \hat{k}

are orthogonal and

|\vec{a}| = |\vec{b}|

, then

(λ, μ) =

HARD

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

Let

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

and

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

be two vectors. Consider a vector

\vec{c} = α \vec{a} + β \vec{b}, α, β \in R .

If the projection of

\vec{c}

on the vector

(\vec{a} + \vec{b})

3 \sqrt{2},

then the minimum value of

(\vec{c} - (\vec{a} \times \vec{b})) . \vec{c}

equals

EASY

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

If the vectors

a \hat{i} + \hat{j} + \hat{k}, \hat{i} + b \hat{j} + \hat{k}

and

\hat{i} + \hat{j} + c \hat{k}

are coplanar

(a \neq b \neq c \neq 1)

, then the value of

a b c - (a + b + c) =

EASY

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

The value of

m,

if the vectors

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

and

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

are coplanar, is

MEDIUM

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

Let

O

be the origin. Let

∴ P Q, P R, P S

and

P, Q, R

be such that

| \vec{P Q} | = \sqrt{20}

and the vector

\vec{O P}

is perpendicular to

\vec{O Q} .

\vec{O R} = 3 \hat{i} + z \hat{j} - 7 \hat{k}, z \in R,

is coplanar with

\vec{O P}

and

\vec{O Q},

then the value of

x^{2} + y^{2} + z^{2}

is equal to

EASY

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

Let

α \in R

and the three vectors

\vec{a} = α \hat{i} + \hat{j} + 3 \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - α \hat{k}

and

\vec{c} = α \hat{i} - 2 \hat{j} + 3 \hat{k} .

Then the set S = {

α : \vec{a}, \vec{b}

and

\vec{c}

are coplanar}

MEDIUM

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

The position vector of $A$ and $B$ are $2 \hat{i} + 2 \hat{j} + \hat{k}$ and $2 \hat{i} + 4 \hat{j} + 4 \hat{k}$ . The length of the internal bisector of $∠ B O A$ of triangle $A O B$ is

MEDIUM

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

Given two vectors

\hat{i} - \hat{j}

and

\hat{i} - 2 \hat{j}

. The unit vector, coplanar with the two given vectors and perpendicular to

(i - j)

MEDIUM

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

Let

A, B, C, D

be the points with position vectors

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

and

4 \hat{i} + 5 \hat{j} + λ \hat{k}

respectively. If the points

A, B, C, D

lie on a plane, then the value of

λ

EASY

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

If vectors

\vec{a_{1}} = x \hat{i} - \hat{j} + \hat{k}

and

\vec{a_{2}} = \hat{i} + y \hat{j} + z \hat{k}

are collinear, then a possible unit vector parallel to the vector

x \hat{i} + y \hat{j} + z \hat{k}

is:

HARD

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

The unit vector which is orthogonal to the vector

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

and is coplanar with vectors

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

and

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

EASY

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

A vector

\vec{a}

has components

3 p

and

1

with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system,

\vec{a}

has components

p + 1

and

\sqrt{10},

then a value of

p

is equal to:

MEDIUM

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

If the points with position vectors

30 \hat{i} + 3 \hat{j}, 20 \hat{i} + 6 \hat{j}

and

5 \hat{i} + b \hat{j}

are collinear, then

b

equal

MEDIUM

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

If a vector

\vec{x}

makes angles with measure

\frac{π}{4}

and

\frac{5 π}{4}

with positive directions of

X

-axis and

Y

-axis respectively, then

\vec{x}

made angle of measure …... with positive direction of

Z

-axis

HARD

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

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

are non-coplaner vectors such that

\vec{b} \times \vec{c} = \vec{a}; \vec{c} \times \vec{a} = \vec{b}; \vec{a} \times \vec{b} = \vec{c}

, then which of the following is not TRUE?

EASY

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

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

and

\vec{c} = x \hat{i} + (x - 2) \hat{j} - \hat{k}

and if the vector

\vec{c}

lies in the plane of vectors

\vec{a}

and

\vec{b},

then

x

equals

EASY

Mathematics>Vectors and 3D Geometry>Addition of Vectors>Types of Vectors

The sum of the distinct real values of

μ

for which the vectors

μ \hat{i} + \hat{j} + \hat{k}, \hat{i} + μ \hat{j} + \hat{k}, \hat{i} + \hat{j} + μ \hat{k}

are co-planar, is

If a¯, b¯, c¯ are non-coplanar vectors, then show that the points with the position vectors 3a¯+4b¯-2c¯, a¯-2b¯+3c¯, a¯-6b¯+6c¯ and 2a¯+3b¯-c¯ are coplanar.

Important Questions on Addition of Vectors

Learn and Prepare for any exam you want !

If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors, then show that the points with the position vectors $3 \bar{a} + 4 \bar{b} - 2 \bar{c}, \bar{a} - 2 \bar{b} + 3 \bar{c}, \bar{a} - 6 \bar{b} + 6 \bar{c}$ and $2 \bar{a} + 3 \bar{b} - \bar{c}$ are coplanar.