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]

Question

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]

Smart Achievers · Accepted Answer

(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

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

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Solution

Q. Let a→ and b→ be two unit vectors such that the angle between them is π3. If λa→+2b→ and 3a→–λb→ 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

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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]