Let and be two unit vectors such that the angle between them is . If and are perpendicular to each other, then the number of values of in [–1, 3] is : [2025]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{π}{3}$ . If $λ \vec{a} + 2 \vec{b}$ and $3 \vec{a} - λ \vec{b}$ are perpendicular to each other, then the number of values of $λ$ in [–1, 3] is : [2025]

1 1

2 0

3 2

4 3

Ans.
(2)

$\vec{a} \cdot \vec{b} =$ $| \vec{a} |$ $| \vec{b} |$ $\cos \frac{π}{3} = \frac{1}{2}$ [ $∵ | \vec{a} | = | \vec{b} | = 1$ ]

Now, $(λ \vec{a} + 2 \vec{b}) \cdot (3 \vec{a} - λ \vec{b}) = 0$

$\Rightarrow 3 λ \vec{a} \cdot \vec{a} - λ^{2} \vec{a} \cdot \vec{b} + 6 \vec{b} \cdot \vec{a} - 2 λ \vec{b} \cdot \vec{b} = 0$

$\Rightarrow 3 λ - \frac{λ^{2}}{2} + 3 - 2 λ = 0$ $[∵ | \vec{a} | = | \vec{b} | = 1 and \vec{a} \cdot \vec{b} = \frac{1}{2}]$

$\Rightarrow λ^{2} - 2 λ - 6 = 0 \Rightarrow λ = 1 \pm \sqrt{7} \notin [- 1, 3]$

$\Rightarrow$ Number of values of $λ$ = 0

Want Us to Email you About Special Offers & Updates?