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Vector Algebra

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Chapter Elite-20 Challenge Zone

Target Advanced

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 2025 (January)

LEVELJEE Advanced

Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be $\hat{i} + 2 \hat{j} + \hat{k},$ $\hat{i} + 3 \hat{j} - 2 \hat{k}$ and $2 \hat{i} + \hat{j} - \hat{k}$ respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E. If the length of AD is $\frac{110}{3}$ and the volume of the tetrahedron is $\frac{805}{6 2},$ then the position vector of E is

(A)

\frac{1}{12} (7 \hat{i} + 4 \hat{j} + 3 \hat{k})

(B)

\frac{1}{2} (\hat{i} + 4 \hat{j} + 7 \hat{k})

(C)

\frac{1}{6} (12 \hat{i} + 12 \hat{j} + \hat{k})

(D)

\frac{1}{6} (7 \hat{i} + 12 \hat{j} + \hat{k})

JEE Main 2023 (25 January Shift 1)

LEVELJEE Advanced

The vector $a = - \hat{i} + 2 \hat{j} + \hat{k}$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $b$ . Then the projection of $3 a + 2 b$ on $c = 5 \hat{i} + 4 \hat{j} + 3 \hat{k}$ is

(A)

32

(B)

(C)

6

(D)

23

JEE Main 2023 (01 February Shift 1)

LEVELJEE Advanced

Let $v = α \hat{i} + 2 \hat{j} - 3 \hat{k}, w = 2 α \hat{i} + \hat{j} - \hat{k}$ , and $u$ be a vector such that $∣ u ∣ = α > 0$ . If the minimum value of the scalar triple product $[u v w]$ is $- α 3401$ , and $∣ u \cdot \hat{i} ∣^{2} = m / n$ where m and n are coprime natural numbers, then $m + n$ is equal to

JEE(ADVANCED)-201

LEVELJEE Advanced

Comprehension Passage

Let O be the origin, and

O X, O Y, O Z

be three unit vectors in the directions of the sides

QR, R P, P Q

, respectively, of a triangle PQR.

Question 1:

$∣ O X \times O Y ∣ =$

(A)

sin (P + Q)

(B)

sin 2 R

(C)

sin (P + R)

(D)

sin (Q + R)

Question 2:

If the triangle PQR varies, then the minimum value of $cos (P + Q) + cos (Q + R) + cos (R + P)$ is

(A)

\frac{5}{3}

(B)

\frac{3}{2}

(C)

- \frac{3}{2}

(D)

- \frac{5}{3}

JEE Advanced 2014

LEVELJEE Advanced

Match List I with List II:

List-I

(P)

Let

y (x) = cos (3 cos^{- 1} x), x \in [- 1, 1], x \neq = \pm \frac{3}{2}

. Then

\frac{1}{y ( x )} {(x^{2} - 1) \frac{d ^{2} y ( x )}{d x ^{2}} + x \frac{d y ( x )}{d x}}

equals

(Q)

Let

A_{1}, A_{2}, \dots, A_{n} (n > 2)

be the vertices of a regular polygon of

n

sides with its centre at the origin. Let

a_{k}

be the position vector of the point

A_{k}, k = 1, 2, \dots, n

. If

∣ \sum_{k = 1}^{n - 1} (a_{k} \times a_{k + 1}) ∣ = ∣ \sum_{k = 1}^{n - 1} (a_{k} \cdot a_{k + 1}) ∣

, then the minimum value of

n

(R)

If the normal from the point

P (h, 1)

on the ellipse

\frac{x ^{2}}{6} + \frac{y ^{2}}{3} = 1

is perpendicular to the line

x + y = 8

, then the value of

h

(S)

Number of positive solutions satisfying the equation

tan^{- 1} (\frac{1}{2 x + 1}) + tan^{- 1} (\frac{1}{4 x + 1}) = tan^{- 1} (\frac{2}{x ^{2}})

List-II

Match the following:

List-I

(P)

R^{2}

, if the magnitude of the projection vector of the vector

α \hat{i} + β \hat{j}

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

3

and if

α = 2 + 3 β

, then possible value of

∣ α ∣

is/are

(Q)

Let

a

and

b

be real numbers such that the function

f (x) = {- 3 a x^{2} - 2, b x + a^{2}, x < 1 x \geq 1

if differentiable for all

x \in R

. Then possible value of

a

is (are)

(R)

Let

ω \neq = 1

be a complex cube root of unity. If

(3 - 3 ω + 2 ω^{2})^{4 n + 3} + (2 + 3 ω - 3 ω^{2})^{4 n + 3} + (- 3 + 2 ω + 3 ω^{2})^{4 n + 3} = 0

, then possible value (s) of

n

is (are)

(S)

Let the harmonic mean of two positive real numbers

a

and

b

be 4. If

q

is a positive real number such that

a, 5, q, b

is an arithmetic progression, then the value(s) of

∣ q - a ∣

is (are)

List-II

From a point $O$ inside a triangle $A B C$ , perpendiculars $O D, O E, O F$ are drawn to the sides $B C, C A, A B$ respectively. Prove that the perpendiculars from $A, B, C$ to the sides $E F, F D, D E$ are concurrent.

JEE Advanced 1999

LEVELJEE Advanced

Let $u$ and $v$ be unit vectors. If $w$ is a vector such that $w + (w \times u) = v$ , then prove that $∣ (u \times v) \cdot w ∣ \leq 1/2$ and that the equality holds if and only if $u$ is perpendicular to $v$ .

JEE Main 2020 - 3 Sep (Evening)

LEVELJEE Advanced

Let $a, b, c \in R$ be such that $a^{2} + b^{2} + c^{2} = 1$ . If $a cos θ = b cos (θ + \frac{2 π}{3}) = c cos (θ + \frac{4 π}{3})$ , where $θ = \frac{π}{9}$ , then the angle between the vectors $a \hat{i} + b \hat{j} + c \hat{k}$ and $b \hat{i} + c \hat{j} + a \hat{k}$ is :

(A)

(B)

\frac{π}{9}

(C)

\frac{π}{2}

(D)

\frac{2 π}{3}

JEE Advanced 2021

LEVELJEE Advanced

Let $O$ be the origin and $O A = 2 \hat{i} + 2 \hat{j} + \hat{k}$ , $O B = \hat{i} - 2 \hat{j} + 2 \hat{k}$ and $O C = \frac{1}{2} (O B - λ O A)$ for some $λ > 0$ . If $∣ O B \times O C ∣ = \frac{9}{2}$ , then which of the following statements is (are) TRUE?

* Multiple Correct Options

(A)

Projection of

O C

O A

- \frac{3}{2}

(B)

Area of the triangle

O A B

\frac{9}{2}

(C)

Area of the triangle

A B C

\frac{9}{2}

(D)

The acute angle between the diagonals of the parallelogram with adjacent sides

O A

and

O C

\frac{π}{3}

JEE Advanced 2020

LEVELJEE Advanced

Let $a$ and $b$ be positive real numbers. Suppose $P Q = a \hat{i} + b \hat{j}$ and $P S = a \hat{i} - b \hat{j}$ are adjacent sides of a parallelogram $P QR S$ . Let $u$ and $v$ be the projection vectors of $w = \hat{i} + \hat{j}$ along $P Q$ and $P S$ , respectively. If $∣ u ∣ + ∣ v ∣ = ∣ w ∣$ and if the area of the parallelogram $P QR S$ is 8, then which of the following statements is/are TRUE?

* Multiple Correct Options

(A)

a + b = 4

(B)

a - b = 2

(C)

P R

of the parallelogram

P QR S

is 4

(D)

w

is an angle bisector of the vectors

P Q

and

P S

JEE Advanced 2001

LEVELJEE Advanced

Let $A (t) = f_{1} (t) \hat{i} + f_{2} (t) \hat{j}$ and $B (t) = g_{1} (t) \hat{i} + g_{2} (t) \hat{j}, t \in [0, 1]$ , where $f_{1}, f_{2}, g_{1}, g_{2}$ are continuous functions. If $A (t)$ and $B (t)$ are nonzero vectors for all $t$ and $A (0) = 2 \hat{i} + 3 \hat{j}, A (1) = 6 \hat{i} + 2 \hat{j}, B (0) = 3 \hat{i} + 2 \hat{j}$ and $B (1) = 2 \hat{i} + 6 \hat{j}$ . Then show that $A (t)$ and $B (t)$ are parallel for some $t$ .

JEE Advanced 2000

LEVELJEE Advanced

Let $A B C$ and $P QR$ be any two triangles in the same plane. Assume that the perpendiculars from the points $A, B, C$ to the sides $QR, R P, P Q$ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from $P, Q, R$ to $B C, C A, A B$ respectively are also concurrent.

JEE Main 2026 (28 January Shift 1)

LEVELJEE Advanced

Let $P QR$ be a triangle such that $P Q = - 2 \hat{i} - \hat{j} + 2 \hat{k}$ and $P R = a \hat{i} + b \hat{j} - 4 \hat{k}, a, b \in Z$ . Let $S$ be the point on $QR$ , which is equidistant from the lines $P Q$ and $P R$ . If $∣ P R ∣ = 9$ and $P S = \hat{i} - 7 \hat{j} + 2 \hat{k}$ , then the value of $3 a - 4 b$ is ......... .

JEE Main 2022 (29 July Shift 1)

LEVELJEE Advanced

Let $\overset{a}{^}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{π}{4}$ . If $θ$ is the angle between the vectors $(\overset{a}{^} + \hat{b})$ and $(\overset{a}{^} + 2 \hat{b} + 2 (\overset{a}{^} \times \hat{b}))$ , then the value of $164 cos^{2} θ$ is equal to :

(A)

90 + 272

(B)

45 + 182

(C)

90 + 32

(D)

54 + 902

JEE Main 2026 (22 January Shift 2)

LEVELJEE Advanced

Let $a = 2 \hat{i} - \hat{j} + \hat{k}$ and $b = λ \hat{j} + 2 \hat{k}, λ \in Z$ be two vectors. Let $c = a \times b$ and $d$ be a vector of magnitude 2 in $y z$ -plane. If $∣ c ∣ = 53$ , then the maximum possible value of $(c \cdot d)^{2}$ is equal to :

JEE Main 2026 (28 January Shift 2)

LEVELJEE Advanced

Let $P$ be a point in the plane of the vectors $A B = 3 \hat{i} + \hat{j} - \hat{k}$ and $A C = \hat{i} - \hat{j} + 3 \hat{k}$ such that $P$ is equidistant from the lines AB and AC. If $∣ A P ∣ = \frac{5}{2}$ , then the area of the triangle ABP is:

(A)

(B)

\frac{30}{4}

(C)

\frac{3}{2}

(D)

\frac{26}{4}

JEE Main 2026 (21 January Shift 1)

LEVELJEE Advanced

Let $c$ and $d$ be vectors such that $∣ c + d ∣ = 29$ and $c \times (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) \times d$ . If $λ_{1}, λ_{2} (λ_{1} > λ_{2})$ are the possible values of $(c + d) \cdot (- 7 \hat{i} + 2 \hat{j} + 3 \hat{k})$ , then the equation $K^{2} x^{2} + (K^{2} - 5 K + λ_{1}) x y + (3 K + \frac{λ _{2}}{2}) y^{2} - 8 x + 12 y + λ_{2} = 0$ represents a circle, for $K$ equal to :

JEE Main 2025 (January)

LEVELJEE Advanced

Let the point A divide the line segment joining the points $P (- 1, - 1, 2)$ and $Q (5, 5, 10)$ internally in the ratio $r : 1 (r > 0)$ If O is the origin and $(O Q \cdot O A) - \frac{1}{5} ∣ O P \times O A ∣^{2} = 10$ then the value of r is:

(A)

7

(B)

(C)

(D)

Micro-Topic Strategy Matrix

Scalar and Vector Quantities

Addition of Vectors

Components of a Vector

Scalar (Dot) Product

Vector (Cross) Product

Scalar Triple Product

Vector Triple Product

Chapter Elite-20 Challenge Zone

Match the following:

List-I

List-II

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣OX×OY∣=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the triangle PQR varies, then the minimum value of cos(P+Q)+cos(Q+R)+cos(R+P) is

Match List I with List II:

List-I

List-II

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 u and v be unit vectors. If w is a vector such that w+(w×u)=v, then prove that ∣(u×v)⋅w∣≤1/2 and that the equality holds if and only if u is perpendicular to v.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c∈R be such that a2+b2+c2=1. If acosθ=bcos(θ+32π​)=ccos(θ+34π​), where θ=9π​, then the angle between the vectors ai^+bj^​+ck^ and bi^+cj^​+ak^ is :

$∣ O X \times O Y ∣ =$

If the triangle PQR varies, then the minimum value of $cos (P + Q) + cos (Q + R) + cos (R + P)$ is

Let $u$ and $v$ be unit vectors. If $w$ is a vector such that $w + (w \times u) = v$ , then prove that $∣ (u \times v) \cdot w ∣ \leq 1/2$ and that the equality holds if and only if $u$ is perpendicular to $v$ .

Let $a, b, c \in R$ be such that $a^{2} + b^{2} + c^{2} = 1$ . If $a cos θ = b cos (θ + \frac{2 π}{3}) = c cos (θ + \frac{4 π}{3})$ , where $θ = \frac{π}{9}$ , then the angle between the vectors $a \hat{i} + b \hat{j} + c \hat{k}$ and $b \hat{i} + c \hat{j} + a \hat{k}$ is :