Three vectors P → , Q → and R → are shown in the figure. Let, S be any point on the vector R → . The distance between the points P and S is b R → . The general relation among vectors P → , Q → and S → are

S → = 1 - b P → + b 2 Q →

S → = 1 - b 2 P → + b Q →

EASY

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If $\vec{A}, \vec{B} & \vec{C}$ represent unit vectors in each case, then which vector combination(s) results in a unit vector $?$

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Important Questions on Vector Algebra

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

3 \hat{i} + 2 \hat{j} - 5 \hat{k} = x (2 \hat{i} - \hat{j} + \hat{k}) + y (\hat{i} + 3 \hat{j} - 2 \hat{k}) + z (- 2 \hat{i} + \hat{j} - 3 \hat{k})

,

then

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition 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:

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

If two forces of $3$ units and $4$ units are acting at an angle $90 °$ , then its resultant forces will be

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

\vec{a} = 2 \hat{ı} + 3 \hat{ȷ} - 4 \hat{k}

and

\vec{b} = \hat{ı} + 3 \hat{ȷ} + 2 \hat{k}

, then a unit vector in the direction of

\vec{a} + \vec{b}

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition 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

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

The two adjacent sides of 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 its diagonal which starts at the common vertex of the above two sides, is

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

\vec{a} = 2 \hat{i} + 5 \hat{j}

and

\vec{b} = 2 \hat{i} - \hat{j}

, then the unit vector along

\vec{a} + \vec{b}

will be

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of 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

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

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

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition 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

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

In quadrilateral

A B C D, \vec{A B} = \vec{a}, \vec{B C} = \vec{b}, \vec{D A} = \vec{a} - \vec{b}, M

is the midpoint of

B C

and

X

is a point on

D M

such that

\vec{D X} = \frac{4}{5} \vec{D M}

. Then the points

A, X

and

C

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition 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

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition 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>Vectors and 3D Geometry>Vector Algebra>Addition 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.

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of 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:

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

Three vectors

\vec{P}, \vec{Q}

and

\vec{R}

are shown in the figure. Let,

S

be any point on the vector

\vec{R}

. The distance between the points

P

and

S

b |\vec{R}|

. The general relation among vectors

\vec{P}, \vec{Q}

and

\vec{S}

are

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of 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

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

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

are position vectors of

A, B, C

respectively and if

D, E

are mid points of sides

B C

and

A C

, then

\vec{D E}

is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition of Vectors

Let

u

and

v

be two vectors. Then,

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

if and only if

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Addition 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}

If A→,B →& C→ represent unit vectors in each case, then which vector combination(s) results in a unit vector?

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

If $\vec{A}, \vec{B} & \vec{C}$ represent unit vectors in each case, then which vector combination(s) results in a unit vector $?$