Let a , b , c be three vectors such that a × b = 2 ( a × c ) . If | a| = 1 , | b | = 4 , | c | = 2 , and the angle between b and c i s 60 ° , t h e n | a · c | is equal to [2026]

Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{a} \times \vec{b} = 2 (\vec{a} \times \vec{c}) .$ If $| \vec{a} |$ $= 1,$ $| \vec{b} |$ $= 4,$ $| \vec{c} |$ $= 2$ , and the angle between $\vec{b} and \vec{c} is 60 °, then$ $| \vec{a} \cdot \vec{c} |$ is equal to [2026]

Smart Achievers · Accepted Answer

(3)

$\vec{a} \times \vec{b} - 2 (\vec{a} \times \vec{c}) = 0$

$\vec{a} \times (\vec{b} - 2 \vec{c}) = 0 \Rightarrow \vec{b} - 2 \vec{c} = λ \vec{a} . . . (1)$

$| λ \vec{a} |^{2}$ $=$ $| \vec{b} - 2 \vec{c} |^{2}$ $\Rightarrow λ^{2} | \vec{a} |^{2} = b^{2} + 4 c^{2} - 4 \vec{b} \cdot \vec{c}$

$λ^{2} = 16 + 16 - 4 \cdot 4 \cdot 2 \cdot \frac{1}{2}$

$λ^{2} = 16$

$λ = \pm 4$

$∵$ $\vec{b} - 2 \vec{c} = \pm 4 \vec{a}$

$Dot with \vec{c} \Rightarrow \vec{b} \cdot \vec{c} - 2 | \vec{c} |^{2} = \pm 4 (\vec{a} \cdot \vec{c})$

$4 \cdot 2 \cdot \frac{1}{2} - 2 \cdot 4 = \pm 4 (\vec{a} \cdot \vec{c})$

$| \vec{a} \cdot \vec{c} |$ $= 1$

Solution

Q.
Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{a} \times \vec{b} = 2 (\vec{a} \times \vec{c}) .$ If $| \vec{a} |$ $= 1,$ $| \vec{b} |$ $= 4,$ $| \vec{c} |$ $= 2$ , and the angle between $\vec{b} and \vec{c} is 60 °, then$ $| \vec{a} \cdot \vec{c} |$ is equal to [2026]

1 2

2 4

3 1

4 0

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Solution

Q. Let a→,b→,c→ be three vectors such that a→×b→=2(a→×c→). If |a→|=1, |b→|=4, |c→|=2, and the angle between b→ and c→ is 60°, then |a→·c→| is equal to [2026]

1 2

2 4

3 1

4 0

Ans. (3) a→×b→-2(a→×c→)=0 a→×(b→-2c→)=0⇒b→-2c→=λa→ ...(1) |λa→|2=|b→-2c→|2⇒λ2|a→|2=b2+4c2-4b→·c→ λ2=16+16-4·4·2·12 λ2=16 λ=±4 ∵ b→-2c→=±4a→ Dot with c→⇒b→·c→-2|c→|2=±4(a→·c→) 4·2·12-2·4=±4(a→·c→) |a→·c→|=1

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Q.
Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{a} \times \vec{b} = 2 (\vec{a} \times \vec{c}) .$ If $| \vec{a} |$ $= 1,$ $| \vec{b} |$ $= 4,$ $| \vec{c} |$ $= 2$ , and the angle between $\vec{b} and \vec{c} is 60 °, then$ $| \vec{a} \cdot \vec{c} |$ is equal to [2026]