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If $\vec{A}, \vec{B}$ and $\vec{C}$ are the unit vectors along the incident ray, reflected ray and outward normal to the reflecting surface, then

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Important Questions on Vector Algebra

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Product 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>Coordinate Geometry>Vector Algebra>Product of Vectors

If the sum of two unit vectors is again a unit vector, then magnitude of their difference is

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

\vec{A} = 3 \hat{i} - 2 \hat{j} + \hat{k}, \vec{B} = \hat{i} - 3 \hat{j} + 5 \hat{k}

and

\vec{C} = 2 \hat{i} + \hat{j} - 4 \hat{k}

form a right angled triangle then out of the following which one is satisfied?

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

If the vectors

\vec{AB} = 3 \hat{i} + 4 \hat{k}

and

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

are the sides of a triangle

A B C,

then the length of the median through

A

is:

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

In a triangle $P Q R$ , let $\vec{a} = \vec{Q R}, \vec{b} = \vec{R P}$ and $\vec{c} = \vec{P Q}$ .

If $|\vec{a}| = 3, |\vec{b}| = 4$ and $\frac{\vec{a} . (\vec{c} - \vec{b})}{\vec{c} . (\vec{a} - \vec{b})} = \frac{|\vec{a}|}{|\vec{a}| + |\vec{b}|}$ , then the value of ${|\vec{a} \times \vec{b}|}^{2}$ is _______

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

If vectors

\vec{A} = \cos ω t \hat{i} + s i n ω t \hat{j}

and

\vec{B} = c o s \frac{ω t}{2} \hat{i} + \sin \frac{ω t}{2} \hat{j}

are functions of time, then the value of

t

at which they are orthogonal to each other is:

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

The diagonals of a parallelogram are the vectors

3 \hat{i} + 6 \hat{j} - 2 \hat{k}

and

- \hat{i} - 2 \hat{j} - 8 \hat{k}

then, the length of the shorter side of parallelogram is

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

The area (in sq. units) of the parallelogram whose diagonals are along the vectors

8 \hat{i} - 6 \hat{j}

and

3 \hat{i} + 4 \hat{j} - 12 \hat{k},

is:

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

In a

∆ A B C,

D, E, F

are the mid-points of the sides

B C

C A

and

A B

respectively, the vector

\vec{A D}

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

Let

u

and

v

be two vectors. Then,

|u - v| = | |u| - |v| |

if and only if

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Product 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>Coordinate Geometry>Vector Algebra>Product of Vectors

Let

\vec{P R} = 3 \hat{i} + \hat{j} - 2 \hat{k} & \vec{S Q} = \hat{i} - 3 \hat{j} - 4 \hat{k}

determine diagonals of a parallelogram

P Q R S

and

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

be another vector. Then the volume (in cubic unit) of the parallelepiped determined by the vectors

\vec{P T}, \vec{P Q} & \vec{P S}

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

Let

\vec{A} = (\hat{i} + \hat{j}) a n d \vec{B} = (2 \hat{i} - \hat{j})

. The magnitude of a coplanar vector

\vec{C}

such that

\vec{A} . \vec{C} = \vec{B} . \vec{C} = \vec{A} . \vec{B}

is given by:

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Product 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:

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

Given, $\vec{a} = 2 \hat{i} + \hat{j} - 2 \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} .$ Let $\vec{c}$ be a vector such that $|\vec{c} - \vec{a}| = 3, |(\vec{a} \times \vec{b}) \times \vec{c}| = 3$ and the angle between $\vec{c}$ and $\vec{a} \times \vec{b}$ be $30 °$ . Then $\vec{a} \cdot \vec{c}$ is equal to:

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

|\vec{a}| = 2, |\vec{b}| = 3

and

|2 \vec{a} - \vec{b}| = 5,

then

|2 \vec{a} + \vec{b}|

equals :

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

If the vector

\vec{b} = 3 \hat{j} + 4 \hat{k}

is written as the sum of a vector

\vec{b_{1}}

, parallel to

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

and a vector

{\vec{b}}_{2},

perpendicular to

\vec{a},

then

{\vec{b}}_{1} \times {\vec{b}}_{2}

is equal to :

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

The two vectors

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

and

\hat{i} + 3 \hat{j} + 5 \hat{k}

represent the two sides

\vec{A B}

and

\vec{A C}

respectively of a

Δ A B C

. The length of the median through

A

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

\vec{a}

and

\vec{b}

are unit vectors, then the greatest value of

\sqrt{3} | \vec{a} + \vec{b} | + | \vec{a} - \vec{b} |

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Product of Vectors

Let

\vec{v}

be a vector in the plane such that

| \vec{v} - \vec{i} | = | \vec{v} - 2 \vec{i} | = | \vec{v} - \vec{j} | .

Then

| \vec{v} |

lies in the interval.

If A → , B → and C → are the unit vectors along the incident ray, reflected ray and outward normal to the reflecting surface, then

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

If $\vec{A}, \vec{B}$ and $\vec{C}$ are the unit vectors along the incident ray, reflected ray and outward normal to the reflecting surface, then