Components of a Vector

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JEE Main 2009

The projections of a vector on the three coordinate axis are $6, - 3, 2$ respectively. The direction cosines of the vector are :

\frac{6}{5}, \frac{- 3}{5}, \frac{2}{5}

(B)

\frac{6}{7}, \frac{- 3}{7}, \frac{2}{7}

(C)

\frac{- 6}{7}, \frac{- 3}{7}, \frac{2}{7}

(D)

6, - 3, 2

JEE Main 2007

LEVELJEE Main

A body weighing 13 kg is suspended by two strings 5m and 12m long, their other ends being fastened to the extremities of a rod 13m long. If the rod be so held that the body hangs immediately below the middle point, then tensions in the strings are

A velocity $\frac{1}{4}$ m/s is resolved into two components along OA and OB making angles $3 0^{\circ}$ and $4 5^{\circ}$ respectively with the given velocity. Then the component along OB is

(A)

\frac{1}{8} (6 - 2)

m/s

(B)

\frac{1}{4} (3 - 1)

m/s

(C)

\frac{1}{4}

m/s

(D)

\frac{1}{8}

m/s

JEE Main 2003

LEVELJEE Main

Consider points A, B, C and D with position vectors $7 \hat{i} - 4 \hat{j} + 7 \hat{k}, \hat{i} - 6 \hat{j} + 10 \hat{k}, - \hat{i} - 3 \hat{j} + 4 \hat{k}$ and $5 \hat{i} - \hat{j} + 5 \hat{k}$ respectively. Then ABCD is a

JEE Advanced 1999

LEVELJEE Main

Let $a = 2 \hat{i} + \hat{j} + \hat{k}, b = \hat{i} + 2 \hat{j} - \hat{k}$ and a unit vector $c$ be coplanar. If $c$ is perpendicular to $a$ , then $c =$

(A)

\frac{1}{2} (- \hat{j} + \hat{k})

(B)

\frac{1}{3} (- \hat{i} - \hat{j} - \hat{k})

(C)

\frac{1}{5} (\hat{i} - 2 \hat{j})

(D)

\frac{1}{3} (\hat{i} - \hat{j} - \hat{k})

JEE Advanced 1994

LEVELJEE Main

Let $α, β, γ$ be distinct real numbers. The points with position vectors $α \hat{i} + β \hat{j} + γ \hat{k}$ , $β \hat{i} + γ \hat{j} + α \hat{k}$ , $γ \hat{i} + α \hat{j} + β \hat{k}$

(A)

are collinear

(B)

form an equilateral triangle

(C)

form a scalene triangle

(D)

form a right angled triangle

JEE Advanced 1986

LEVELJEE Main

A vector $a$ has components $2 p$ 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 the new system, $a$ has components $p + 1$ and 1, then

(A)

p = 0

(B)

p = 1

p = - 1/3

(C)

p = - 1

p = 1/3

(D)

p = 1

p = - 1

The points with position vectors $60 \hat{i} + 3 \hat{j}, 40 \hat{i} - 8 \hat{j}, a \hat{i} - 52 \hat{j}$ are collinear if

(A)

a = - 40

(B)

a = 40

(C)

a = 20

Suppose that $p, q$ and $r$ are three non-coplanar vectors in $R^{3}$ . Let the components of a vector $s$ along $p, q$ and $r$ be 4, 3 and 5, respectively. If the components of this vector $s$ along $(- p + q + r), (p - q + r)$ and $(- p - q + r)$ are $x, y$ and $z$ , respectively, then the value of $2 x + y + z$ is .........

JEE Advanced 2001

LEVELJEE Main

Find 3-dimensional vectors $v_{1}, v_{2}, v_{3}$ satisfying $v_{1} \cdot v_{1} = 4, v_{1} \cdot v_{2} = - 2, v_{1} \cdot v_{3} = 6, v_{2} \cdot v_{2} = 2, v_{2} \cdot v_{3} = - 5, v_{3} \cdot v_{3} = 29$ .

JEE Advanced 1996

LEVELJEE Main

If $b$ and $c$ are any two non-collinear unit vectors and $a$ is any vector, then $(a \cdot b) b + (a \cdot c) c + \frac{a \cdot ( b \times c )}{∣ b \times c ∣ ^{2}} (b \times c) = .........$

JEE Advanced 1993

LEVELJEE Main

In a triangle $A B C$ , $D$ and $E$ are points on $B C$ and $A C$ respectively, such that $B D = 2 D C$ and $A E = 3 E C$ . Let $P$ be the point of intersection of $A D$ and $B E$ . Find $B P / P E$ using vector methods.

JEE Advanced 1989

LEVELJEE Main

In a triangle $O A B$ , $E$ is the midpoint of $B O$ and $D$ is a point on $A B$ such that $A D : D B = 2 : 1$ . If $O D$ and $A E$ intersect at $P$ , determine the ratio $O P : P D$ using vector methods.

JEE Advanced 1988

LEVELJEE Main

The components of a vector $a$ along and perpendicular to a non-zero vector $b$ are ......... and ......... respectively.

JEE Advanced 1983

LEVELJEE Main

A vector $A$ has components $A_{1}, A_{2}, A_{3}$ in a right-handed rectangular Cartesian coordinate system $o x y z$ . The coordinate system is rotated about the $x$ -axis through an angle $\frac{π}{2}$ . Find the components of $A$ in the new coordinate system, in terms of $A_{1}, A_{2}, A_{3}$ .

JEE Advanced 2015

LEVELJEE Advanced

Match the following:

List-I

(P)

In a triangle

Δ X Y Z

, let

a, b

, and

c

be the lengths of the sides opposite to the angles

X, Y

and

Z

, respectively. If

2 (a^{2} - b^{2}) = c^{2}

and

λ = \frac{s i n ( X - Y )}{s i n Z}

, then possible values of

n

for which

cos (nπ λ) = 0

is (are)

(Q)

In a triangle

Δ X Y Z

, let

a, b

and

c

be the lengths of the sides opposite to the angles

X, Y

, and

Z

respectively. If

1 + cos 2 X - 2 cos 2 Y = 2 sin X sin Y

, then possible value(s) of

\frac{a}{b}

is (are)

(R)

R^{2}

, let

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

and

β \hat{i} + (1 - β) \hat{j}

be the position vectors of

X, Y

and

Z

with respect to the origin

O

, respectively. If the distance of

Z

from the bisector of the acute angle of

O X

with

O Y

\frac{3}{2}

, then possible value(s) of

∣ β ∣

is (are)

(S)

Suppose that

F (α)

denotes the area of the region bounded by

x = 0, x = 2, y^{2} = 4 x

and

y = ∣ α x - 1∣ + ∣ α x - 2∣ + α x

, where

α \in {0, 1}

. Then the value(s) of

F (α) + \frac{8}{3} 2

, when

α = 0

and

α = 1

, is (are)

List-II

JEE Main 2019 (10 January)

LEVELJEE Main

Let $α = (λ - 2) a + b$ and $β = (4 λ - 2) a + 3 b$ be two given vectors where vectors $a$ and $b$ are non-collinear. The value of $λ$ for which vectors $α$ and $β$ are collinear, is :

JEE Main 2019 (9 April)

LEVELJEE Main

If a unit vector $a$ makes angles $π /3$ with $\hat{i}$ , $π /4$ with $\hat{j}$ and $θ \in (0, π)$ with $\hat{k}$ , then a value of $θ$ is :-

(A)

5 π /12

(B)

5 π /6

(C)

2 π /3

(D)

π /4

JEE Main 11 Jan 2019 (Evening)

LEVELJEE Main

Let $3 \hat{i} + \hat{j}$ , $\hat{i} + 3 \hat{j}$ and $β \hat{i} + (1 - β) \hat{j}$ respectively be the position vectors of the points $A$ , $B$ and $C$ with respect to the origin $O$ . If the distance of $C$ from the bisector of the acute angle between $O A$ and $O B$ is $\frac{3}{2}$ , then the sum of all possible values of $β$ is:

JEE Main 2020 (7 January Shift 1)

LEVELJEE Main

A vector $a = α \hat{i} + 2 \hat{j} + β \hat{k} (α, β \in R)$ lies in the plane of the vectors, $b = \hat{i} + \hat{j}$ and $c = \hat{i} - \hat{j} + 4 \hat{k}$ . If $a$ bisects the angle between $b$ and $c$ , then

(A)

a . \hat{i} + 3 = 0

(B)

a . \hat{k} + 4 = 0

(C)

a . \hat{i} + 1 = 0

(D)

a . \hat{k} + 2 = 0

JEE Main 2021 (26 February Shift 2)

LEVELJEE Main

If vectors $a_{1} = x \hat{i} - \hat{j} + \hat{k}$ and $a_{2} = \hat{i} + y \hat{j} + z \hat{k}$ are collinear, then a possible unit vector parallel to the vector $x \hat{i} + y \hat{j} + z \hat{k}$ is :

(A)

\frac{1}{2} (- \hat{j} + \hat{k})

(B)

\frac{1}{2} (\hat{i} - \hat{j})

(C)

\frac{1}{3} (\hat{i} - \hat{j} + \hat{k})

(D)

\frac{1}{3} (\hat{i} + \hat{j} - \hat{k})

JEE Main 2021 (March)

LEVELJEE Main

A vector $a$ 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, $a$ has components $p + 1$ and $10$ , then a value of $p$ is equal to:

(A)

(B)

- \frac{5}{4}

(C)

\frac{5}{4}

(D)

-1

JEE Main 2022 (29 June Shift 2)

LEVELJEE Main

Let $A, B, C$ be three points whose position vectors respectively are: $a = \hat{i} + 4 \hat{j} + 3 \hat{k}$ , $b = 2 \hat{i} + α \hat{j} + 4 \hat{k}, α \in R$ , $c = 3 \hat{i} - \2 \hat{j} + 5 \hat{k}$ . If $α$ is the smallest positive integer for which $a, b, c$ are non-collinear, then the length of the median, in $△ A B C$ , through $A$ is:

(A)

\frac{82}{2}

(B)

\frac{62}{2}

(C)

\frac{69}{2}

(D)

\frac{66}{2}

JEE Main 2023 (11 April Shift 1)

LEVELJEE Main

For any vector $a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ , with $10∣ a_{i} ∣ < 1, i = 1, 2, 3$ , consider the following statements: (A) $max {∣ a_{1} ∣, ∣ a_{2} ∣, ∣ a_{3} ∣} \leq ∣ a ∣$ , (B) $∣ a ∣ \leq 3 max {∣ a_{1} ∣, ∣ a_{2} ∣, ∣ a_{3} ∣}$

Both (A) and (B) are true

(D)

Neither (A) nor (B) is true

JEE Main 2023 (24 January Shift 2)

LEVELJEE Main

Let $α = 4 \hat{i} + 3 \hat{j} + 5 \hat{k}$ and $β = \hat{i} + 2 \hat{j} - 4 \hat{k}$ . Let $β_{1}$ be parallel to $α$ and $β_{2}$ be perpendicular to $α$ . If $β = β_{1} + β_{2}$ , then the value of $5 β_{2} \cdot (\hat{i} + \hat{j} + \hat{k})$ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The projections of a vector on the three coordinate axis are 6,−3,2 respectively. The direction cosines of the vector are :

A body weighing 13 kg is suspended by two strings 5m and 12m long, their other ends being fastened to the extremities of a rod 13m long. If the rod be so held that the body hangs immediately below the middle point, then tensions in the strings are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A velocity 41​ m/s is resolved into two components along OA and OB making angles 30∘ and 45∘ respectively with the given velocity. Then the component along OB is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider points A, B, C and D with position vectors 7i^−4j^​+7k^,i^−6j^​+10k^,−i^−3j^​+4k^ and 5i^−j^​+5k^ respectively. Then ABCD is a

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=2i^+j^​+k^,b=i^+2j^​−k^ and a unit vector c be coplanar. If c is perpendicular to a, then c=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β,γ be distinct real numbers. The points with position vectors αi^+βj^​+γk^, βi^+γj^​+αk^, γi^+αj^​+βk^

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The points with position vectors 60i^+3j^​,40i^−8j^​,ai^−52j^​ are collinear if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find 3-dimensional vectors v1​,v2​,v3​ satisfying v1​⋅v1​=4,v1​⋅v2​=−2,v1​⋅v3​=6,v2​⋅v2​=2,v2​⋅v3​=−5,v3​⋅v3​=29.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If b and c are any two non-collinear unit vectors and a is any vector, then (a⋅b)b+(a⋅c)c+∣b×c∣2a⋅(b×c)​(b×c)=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle OAB, E is the midpoint of BO and D is a point on AB such that AD:DB=2:1. If OD and AE intersect at P, determine the ratio OP:PD using vector methods.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The components of a vector a along and perpendicular to a non-zero vector b are ......... and ......... respectively.

Match the following:

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α=(λ−2)a+b and β​=(4λ−2)a+3b be two given vectors where vectors a and b are non-collinear. The value of λ for which vectors α and β​ are collinear, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a unit vector a makes angles π/3 with i^, π/4 with j^​ and θ∈(0,π) with k^, then a value of θ is :-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A vector a=αi^+2j^​+βk^(α,β∈R) lies in the plane of the vectors, b=i^+j^​ and c=i^−j^​+4k^. If a bisects the angle between b and c, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If vectors a1​=xi^−j^​+k^ and a2​=i^+yj^​+zk^ are collinear, then a possible unit vector parallel to the vector xi^+yj^​+zk^ is :

The projections of a vector on the three coordinate axis are $6, - 3, 2$ respectively. The direction cosines of the vector are :

A velocity $\frac{1}{4}$ m/s is resolved into two components along OA and OB making angles $3 0^{\circ}$ and $4 5^{\circ}$ respectively with the given velocity. Then the component along OB is

Consider points A, B, C and D with position vectors $7 \hat{i} - 4 \hat{j} + 7 \hat{k}, \hat{i} - 6 \hat{j} + 10 \hat{k}, - \hat{i} - 3 \hat{j} + 4 \hat{k}$ and $5 \hat{i} - \hat{j} + 5 \hat{k}$ respectively. Then ABCD is a

Let $a = 2 \hat{i} + \hat{j} + \hat{k}, b = \hat{i} + 2 \hat{j} - \hat{k}$ and a unit vector $c$ be coplanar. If $c$ is perpendicular to $a$ , then $c =$

Let $α, β, γ$ be distinct real numbers. The points with position vectors $α \hat{i} + β \hat{j} + γ \hat{k}$ , $β \hat{i} + γ \hat{j} + α \hat{k}$ , $γ \hat{i} + α \hat{j} + β \hat{k}$

The points with position vectors $60 \hat{i} + 3 \hat{j}, 40 \hat{i} - 8 \hat{j}, a \hat{i} - 52 \hat{j}$ are collinear if

Find 3-dimensional vectors $v_{1}, v_{2}, v_{3}$ satisfying $v_{1} \cdot v_{1} = 4, v_{1} \cdot v_{2} = - 2, v_{1} \cdot v_{3} = 6, v_{2} \cdot v_{2} = 2, v_{2} \cdot v_{3} = - 5, v_{3} \cdot v_{3} = 29$ .

If $b$ and $c$ are any two non-collinear unit vectors and $a$ is any vector, then $(a \cdot b) b + (a \cdot c) c + \frac{a \cdot ( b \times c )}{∣ b \times c ∣ ^{2}} (b \times c) = .........$

In a triangle $O A B$ , $E$ is the midpoint of $B O$ and $D$ is a point on $A B$ such that $A D : D B = 2 : 1$ . If $O D$ and $A E$ intersect at $P$ , determine the ratio $O P : P D$ using vector methods.

The components of a vector $a$ along and perpendicular to a non-zero vector $b$ are ......... and ......... respectively.

Let $α = (λ - 2) a + b$ and $β = (4 λ - 2) a + 3 b$ be two given vectors where vectors $a$ and $b$ are non-collinear. The value of $λ$ for which vectors $α$ and $β$ are collinear, is :

If a unit vector $a$ makes angles $π /3$ with $\hat{i}$ , $π /4$ with $\hat{j}$ and $θ \in (0, π)$ with $\hat{k}$ , then a value of $θ$ is :-

A vector $a = α \hat{i} + 2 \hat{j} + β \hat{k} (α, β \in R)$ lies in the plane of the vectors, $b = \hat{i} + \hat{j}$ and $c = \hat{i} - \hat{j} + 4 \hat{k}$ . If $a$ bisects the angle between $b$ and $c$ , then

If vectors $a_{1} = x \hat{i} - \hat{j} + \hat{k}$ and $a_{2} = \hat{i} + y \hat{j} + z \hat{k}$ are collinear, then a possible unit vector parallel to the vector $x \hat{i} + y \hat{j} + z \hat{k}$ is :