Let u ^ = u 1 i ^ + u 2 j ^ + u 3 k ^ be a unit vector in R 3 and w ^ = 1 6 i ^ + j ^ + 2 k ^ . Given that there exists a vector v → in R 3 such that u ^ × v → = 1 and w ^ . u ^ × v → = 1 . Which of the following statement(s) is (are) correct ?

If u ^ lies in the xy - plane then u 1 = u 2

There is exactly one choice for such v →

If u ^ lies in the xz - plane then 2 u 1 = u 3

r → = s i ^ - t k ^ is the equation of :

a straight line joining the points i → and k →

The position vectors of two points A and B are OA ¯ = 2 i ^ - j ^ - k ^ and OB ¯ = 2 i ^ - j ^ + 2 k ^ , respectively. The position vector of a point P which divides the line segment joining A and B in the ratio 2 : 1 is

The position vectors of two points A and B are OA ¯ = 2 i ^ - j ^ - k ^ and OB ¯ = 2 i ^ - j ^ + 2 k ^ , respectively. Let The position vector of a point P which divides the line segment joining A and B in the ratio 2 : 1 is O P ¯ When a point whose position vector is O P ¯ divides the line segment joining the position vectors O A ¯ and O B ¯ in the ratio of m : n , then O P ¯ = n O A ¯ + m O B ¯ m + n ⇒ O P ¯ = O A ¯ + 2 O B ¯ 2 + 1 ⇒ O P ¯ = 2 i ^ - j ^ - k ^ + 2 2 i ^ - j ^ + 2 k ^ 3 ⇒ O P ¯ = 2 i ^ - j ^ + k ^ Thus the required position vector is 2 i ^ - j ^ + k ^ .

MEDIUM

Earn 100

Using vector method, prove that a quadrilateral is a rhombus if and only if diagonals bisect each other at right angles.

Important Questions on Vector Algebra

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Let $\vec{a}, \vec{b} and \vec{c}$ be three non - zero vectors such that no two of them are collinear and $(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3} |\vec{b}| |\vec{c}| \vec{a}$ . If $θ$ is the angle between vectors $\vec{b} and \vec{c}$ , then a value of $\sin θ$ is

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Prove by vector method that the internal bisectors of the angles of a triangle are concurrent.

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Let

P Q R

be a triangle. The points

A, B

and

C

are on the sides

Q R, R P

and

P Q

respectively such that

\frac{Q A}{A R} = \frac{R B}{B P} = \frac{P C}{C Q} = \frac{1}{2}

. Then

\frac{Area (∆ P Q R)}{Area (∆ A B C)}

is equal to

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Let

\vec{α} = (λ - 2) \vec{a} + \vec{b}

and

\vec{β} = (4 λ - 2) \vec{a} + 3 \vec{b},

be two given vectors where vectors

\vec{a}

and

\vec{b}

are non-collinear. The value of

λ

for which vectors

\vec{α}

and

\vec{β}

are collinear, is:

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

In a triangle

A B C

, let

G

denote its centroid and let

M, N

be points in the interiors of the segments

A B, A C

, respectively, such that

M, G, N,

are collinear. If

r

denotes the ratio of the area of triangle

A M N

to the area of

A B C

then

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Let

\vec{a} = 2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k}, \vec{b} = 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k}

and

\vec{c} = 3 \hat{i} + 6 \hat{j} + (λ_{3} - 1) \hat{k}

be three vectors such that

\vec{b} = 2 \vec{a}

and

\vec{a}

is perpendicular to

\vec{c} .

Then a possible value of

(λ_{1}, λ_{2}, λ_{3})

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Let

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

be the position vectors of the vertices of a triangle

A B C

. Through the vertices, lines are drawn parallel to the sides to form the triangle

A^{'} B^{'} C^{'}

. Then the centroid of

Δ A^{'} B^{'} C^{'}

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Let

\vec{a}

and

\vec{b}

be two unit vectors such that

\vec{a} . \vec{b} = 0 .

For some

x, y \in R,

let

\vec{c} = x \vec{a} + y \vec{b} + (\vec{a} \times \vec{b}) .

|\vec{c}| = 2

and the vector

\vec{c}

is inclined at the same angle

α

to both

\vec{a}

and

\vec{b},

then the value of

8 \cos^{2} α

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

△ A B C

is right-angled at

B

where

A (5, 6, 4), B (4, 4, 1) and C (8, 2, x)

, then find the value of

x

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

A (\bar{a})

and

B (\bar{b})

are any two points in the space and

R (\bar{r})

be a point on the line segment

A B

dividing it internally in the ratio

m : n

, then prove that

\bar{r} = \frac{m \bar{b} + n \bar{a}}{m + n}

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Let

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

\vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + \sqrt{2} \hat{k}

and

\vec{c} = 5 \hat{i} + \hat{j} + \sqrt{2} \hat{k}

be three vectors such that the projection vector of

\vec{b}

\vec{a}

|\vec{a}|

. If

\vec{a} + \vec{b}

is perpendicular to

\vec{c}

, then

|\vec{b}|

is equal to:

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

D, E

and

F

are respectively mid-points of

A B, A C

and

B C

∆ A B C

, then

\vec{B E} + \vec{A F}

is equal to

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Let

\vec{a}, \vec{b}

and

\vec{c}

be three unit vectors, out of which vectors

\vec{b}

and

\vec{c}

are non-parallel. If

α

and

β

are the angles which vector

\vec{a}

makes with vectors

\vec{b}

and

\vec{c}

respectively and

\vec{a} \times (\vec{b} \times \vec{c}) = \frac{1}{2} \vec{b}

, then

|α - β|

is equal to :

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Let

\hat{u} = u_{1} \hat{i} + u_{2} \hat{j} + u_{3} \hat{k}

be a unit vector in

R^{3}

and

\hat{w} = \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2 \hat{k}) .

Given that there exists a vector

\vec{v}

R^{3}

such that

|\hat{u} \times \vec{v}| = 1

and

\hat{w} . (\hat{u} \times \vec{v}) = 1 .

Which of the following statement(s) is (are) correct ?

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

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

and

\vec{c} = \hat{i} + m \hat{j} + n \hat{k}

are three coplanar vectors and

|\vec{c}| = \sqrt{6},

then which one of the following is correct?

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

In a parallelogram

A B C D, |\vec{A B}| = a, |\vec{A D}| = b & |\vec{A C}| = c

\vec{D B} \cdot \vec{A B}

has the value:

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

Consider the cube in the first octant with sides

O P, O Q

and

O R

of length

1

, along the

x

-axis,

y

-axis and

z

-axis, respectively, where

O (0, 0, 0)

is the origin. Let

S (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})

be the centre of the cube and

T

be the vertex of the cube opposite to the origin

O

such that

S

lies on the diagonal

O T

. If

\vec{p} = \vec{S P}, \vec{q} = \vec{S Q}, \vec{r} = \vec{S R}

and

\vec{t} = \vec{S T}

, then the value of

|(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t})|

is ______.

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

\vec{r} = s \hat{i} - t \hat{k}

is the equation of :

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

The position vectors of two points

A

and

B

are

\bar{OA} = 2 \hat{i} - \hat{j} - \hat{k}

and

\bar{OB} = 2 \hat{i} - \hat{j} + 2 \hat{k}

, respectively. The position vector of a point

P

which divides the line segment joining

A

and

B

in the ratio

2 : 1

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Geometrical Application of Vectors

\vec{A B} = 3 \hat{ı} + 5 \hat{j} + 4 \hat{k}

\vec{A C} = 5 \hat{i} - 5 \hat{j} + 2 \hat{k}

represent the sides of a triangle

A B C

, then the length of the median through

A

Using vector method, prove that a quadrilateral is a rhombus if and only if diagonals bisect each other at right angles.

Important Questions on Vector Algebra

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