Let a = - i + j + k , b = 2 j + k and vector c satisfying conditions ( i ) [ a b c ] = 0 ( i i ) b · c = a · c = 7 . Then the value of 2 7 | c | 2 is equal to

Question

Let $\vec{a} = - \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} + \hat{k}$ and vector $\vec{c}$ satisfying conditions $(i) [\vec{a} \vec{b} \vec{c}] = 0$ $(i i) \vec{b} \cdot \vec{c} = \vec{a} \cdot \vec{c} = 7 .$ Then the value of $\frac{2}{7}$ $| \vec{c} |^{2}$ is equal to

Smart Achievers · Accepted Answer

(5)

$Let \vec{c} = λ \vec{a} + μ \vec{b}$

$Taking dot by \vec{b}$

$0 = λ (\vec{a} \cdot \vec{b}) + {(\vec{b})}^{2} = - λ + 5 μ$ $\Rightarrow λ - 5 μ = 0 \dots \dots (1)$

$Again \vec{a} \cdot \vec{c} = 7$ $\Rightarrow λ {\vec{a}}^{2} + μ (\vec{a} \cdot \vec{b}) = 7$ $\Rightarrow 3 λ - μ = 7 \dots \dots (2)$

$Solving (1) and (2), λ = \frac{5}{2}, μ = \frac{1}{2}$ $\Rightarrow \frac{2}{7}$ $| \vec{c} |^{2}$ $= \frac{1}{7} \times 35 = 5$

Solution

Q.
Let $\vec{a} = - \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} + \hat{k}$ and vector $\vec{c}$ satisfying conditions $(i) [\vec{a} \vec{b} \vec{c}] = 0$ $(i i) \vec{b} \cdot \vec{c} = \vec{a} \cdot \vec{c} = 7 .$ Then the value of $\frac{2}{7}$ $| \vec{c} |^{2}$ is equal to

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let a→=-i^+j^+k^, b→=2i^+k^ and vector c→ satisfying conditions (i) [a→ b→ c→]=0 (ii) b→·c→=a→·c→=7. Then the value of 27|c→|2 is equal to

Ans. (5) Let c→=λa→+μb→ Taking dot by b→ 0=λ(a→·b→)+(b→)2=-λ+5μ⇒ λ-5μ=0……(1) Again a→·c→=7⇒ λa→ 2+μ(a→·b→)=7⇒ 3λ-μ=7……(2) Solving (1) and (2), λ=52, μ=12⇒ 27|c→|2=17×35=5

Want Us to Email you About Special Offers & Updates?

Q.
Let $\vec{a} = - \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} + \hat{k}$ and vector $\vec{c}$ satisfying conditions $(i) [\vec{a} \vec{b} \vec{c}] = 0$ $(i i) \vec{b} \cdot \vec{c} = \vec{a} \cdot \vec{c} = 7 .$ Then the value of $\frac{2}{7}$ $| \vec{c} |^{2}$ is equal to