Find the magnitude of vector a → = 2 i ^ - 7 j ^ - 3 k ^ .

Given that, the vector is a → = 2 i ^ - 7 j ^ - 3 k ^ We know that, for a vector a → = x i ^ + y j ^ + z k ^ , the magnitude is a ⇀ = x 2 + y 2 + z 2 ⇒ a ⇀ = 2 2 + 7 2 + 3 2 = 4 + 49 + 9 = 62 Thus the magnitude of the given vector is 62

If a → = 3 i ^ + 2 j ^ - 4 k ^ , and b → = i ^ + 2 j ^ + 3 k ^ then find 2 a → + b → , a → · b → , a → × b → , and | a → - b → | .

Given, a → = 3 i ^ + 2 j ^ - 4 k ^ , and b → = i ^ + 2 j ^ + 3 k ^ Then 2 a → + b → = 2 3 i ^ + 2 j ^ - 4 k ^ + i ^ + 2 j ^ + 3 k ^ = 6 i ^ + 4 j ^ - 8 k ^ + i ^ + 2 j ^ + 3 k ^ = 6 + 1 i ^ + 4 + 2 j ^ + - 8 + 3 k ^ = 7 i ^ + 6 j ^ - 5 k ^ Now, a → · b → = ( 3 i ^ + 2 j ^ - 4 k ^ ) · ( i ^ + 2 j ^ + 3 k ^ ) = 3 × 1 + 2 × 2 + ( - 4 ) × 3 = 3 + 4 - 12 = - 5 Now, a → × b → = ( 3 i ^ + 2 j ^ - 4 k ^ ) × ( i ^ + 2 j ^ + 3 k ^ ) = i ^ j ^ k ^ 3 2 - 4 1 2 3 = ( 6 + 8 ) i ^ - ( 9 + 4 ) j ^ + ( 6 - 2 ) k ^ = 14 i ^ - 13 j ^ + 4 k ^ Now, a → - b → = ( 3 i ^ + 2 j ^ - 4 k ^ ) - ( i ^ + 2 j ^ + 3 k ^ ) = ( 3 - 1 i ^ + 2 - 2 j ^ + - 4 - 3 k ^ ) = 2 i ^ - 7 k ^ Hence, a → - b → = 2 2 + - 7 2 = 4 + 49 = 53 units, 2 a → + b → = 7 i ^ + 6 j ^ - 5 k ^ , a → · b → = - 5 , and = 14 i ^ - 13 j ^ + 4 k ^

The projections of a line on the axes are 3 , 4 and 2 6 . Find the length of the line.

Given that, projections of line on axes are 3 , 4 and 2 6 Let d l = 3 , d m = 4 and d n = 2 6 , where d is the length of the line and l , m , n are direction cosines of the line. d 2 l 2 + d 2 m 2 + d 2 n 2 = 9 + 16 + 24 ⇒ d 2 l 2 + m 2 + n 2 = 49 ⇒ d 2 = 49 ⇒ d = 7 ∴ Length of line = 7 units

If a → = 3 i → - j → - 4 k → , b → = - 2 i → + 4 j → - 3 k → then find the magnitude of vector 3 a → - 2 b → .

Given that, the vectors a → = 3 i → - j → - 4 k → , b → = - 2 i → + 4 j → - 3 k → . Consider, 3 a → - 2 b → = 3 3 i → - j → - 4 k → - 2 - 2 i → + 4 j → - 3 k → = 9 i → - 3 j → - 12 k → + 4 i → - 8 j → + 6 k → = 13 i → - 11 j → - 6 k → Magnitude of a vector x i ^ + y j ^ + z k ^ is x 2 + y 2 + z 2 Magnitude of the required vector = 13 2 + 11 2 + 6 2 = 326

HARD

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If $\vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]}, \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$ , where $\vec{a}, \vec{b}$ and $\vec{c}$ are any three vectors forming a right-handed system, then which of the following options are correct (where, $x$ is any positive real number)?

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Important Questions on Vector Algebra

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

If $\vec{a} + \vec{b} + \vec{c} = 0, |\vec{a}| = 3, |\vec{b}| = 4, |\vec{c}| = 5$ then, the angle between $\vec{a}$ and $\vec{b}$ is :

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

The point of intersection of the lines joining points

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

and

- \hat{i}, 2 \hat{i}

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

Find the magnitude of vector

\vec{a} = 2 \hat{i} - 7 \hat{j} - 3 \hat{k}

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

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

then

|\vec{a}|

is :

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

\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) =

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

\vec{a} = 3 \hat{i} + 2 \hat{j} - 4 \hat{k}

, and

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

then find

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

\vec{a} \cdot \vec{b}, \vec{a} \times \vec{b},

and

| \vec{a} - \vec{b} | .

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

The projections of a line on the axes are $3, 4$ and $2 \sqrt{6}$ . Find the length of the line.

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

p \hat{i} + 3 \hat{j}

is a vector of magnitude

5

, then the value of

p

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

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:

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

What is the vector

r

of magnitude

2 \sqrt{3}

units that makes an angle of

\frac{π}{2}

and $\frac{\pi}{6}$ with

y

-axis and

z

-axis respectively?

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

\vec{a} = 3 \vec{i} - \vec{j} - 4 \vec{k}, \vec{b} = - 2 \vec{i} + 4 \vec{j} - 3 \vec{k}

then find the magnitude of vector

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

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

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

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

Work is a vector quantity.

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

Magnitude of the projection of

(- 1, 2, - 1)

\hat{i}

is_____

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

Find the vector Equation and Cartesian Equation of the sphere whose centre is

(- 1, 0, 1)

and radius is

2

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

For any vector

\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}

, with

10 |a_{i}| < 1, i = 1, 2, 3

, consider the following statements:

(A)

\max \{|a_{1}|, |a_{2}|, |a_{3}|\} \leq |\vec{a}|

(B)

| \vec{a} | \leq 3 \max \{|a_{1}|, |a_{2}|, |a_{3}|\}

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

O A B C

is a tetrahedron. If

D, E

are the midpoints of

O A

and

B C

respectively, then

\vec{D E} =

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

Let

\vec{u} = 2 \hat{i} + \hat{j}

and

\vec{v} = 3 \hat{i} - 5 \hat{j}

. Consider three points

P, Q

and

R

having the position vectors

(\frac{5}{2}) \hat{i} - 2 \hat{j}, (\frac{7}{3}) \hat{i} - \hat{j}

and

(\frac{9}{4}) \hat{i}

respectively. Among these, the points in the line passing through

\vec{u}

and

\vec{v}

are

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

P Q R S T

is a pentagon, then the resultant of forces

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

and

\vec{P S}

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Triple Products

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

If p→=b→×c→a→ b→ c→, q→=c→×a→a→ b→ c→, r→=a→×b→a→ b→ c→, where a→,b→ and c→ are any three vectors forming a right-handed system, then which of the following options are correct (where, x is any positive real number)?

Important Questions on Vector Algebra

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