Let are three vectors of which every pair is non collinear, and the vectors and are collinear with and respectively. If , then find

Question

Let $\vec{a}, \vec{b}, \vec{c}$ are three vectors of which every pair is non collinear, and the vectors $\vec{a} + 3 \vec{b}$ and $2 \vec{b} + 3 \vec{c}$ are collinear with $\vec{c}$ and $\vec{a}$ respectively. If $\vec{b} \cdot \vec{b} = 1$ , then find $| 2 \vec{a} + 9 \vec{c} |$ .

Smart Achievers · Accepted Answer

(6)

$Given$

$\vec{a} + 3 \vec{b} = λ \vec{c}$

$2 \vec{b} + 3 \vec{c} = μ \vec{a}$

$\vec{a}$

$\Rightarrow (2 + 3 μ) \vec{b} + (3 - μ λ) \vec{c} = 0$

$\Rightarrow μ = - \frac{2}{3}$

$Thus, 2 \vec{a} + 6 \vec{b} + 9 \vec{c} = \vec{0}$

$\Rightarrow$ $| 2 \vec{a} - 9 \vec{c} |$ $=$ $| 6 \vec{b} |$ $= 6$

Solution

Q.
Let $\vec{a}, \vec{b}, \vec{c}$ are three vectors of which every pair is non collinear, and the vectors $\vec{a} + 3 \vec{b}$ and $2 \vec{b} + 3 \vec{c}$ are collinear with $\vec{c}$ and $\vec{a}$ respectively. If $\vec{b} \cdot \vec{b} = 1$ , then find $| 2 \vec{a} + 9 \vec{c} |$ .

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Solution

Q. Let a→,b→,c→ are three vectors of which every pair is non collinear, and the vectors a→+3b→ and 2b→+3c→ are collinear with c→ and a→ respectively. If b→·b→=1, then find |2a→+9c→|.

Ans. (6) Given a→+3b→=λc→ 2b→+3c→=μa→ ⇒2b→+3c→=μ(λc→-3b→) ⇒(2+3μ)b→+(3-μλ)c→=0 ⇒μ=-23 Thus, 2a→+6b→+9c→=0→ ⇒|2a→-9c→|=|6b→|=6

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Q.
Let $\vec{a}, \vec{b}, \vec{c}$ are three vectors of which every pair is non collinear, and the vectors $\vec{a} + 3 \vec{b}$ and $2 \vec{b} + 3 \vec{c}$ are collinear with $\vec{c}$ and $\vec{a}$ respectively. If $\vec{b} \cdot \vec{b} = 1$ , then find $| 2 \vec{a} + 9 \vec{c} |$ .