ધારો કે vec{a}=hat{i}-alpha hat{j}+beta hat{k | vedclass

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(C) આપેલ સદિશો $\vec{a}=(1, -\alpha, \beta)$,$\vec{b}=(3, \beta, -\alpha)$,અને $\vec{c}=(-\alpha, -2, 1)$ છે,જ્યાં $\alpha, \beta \in \mathbb{Z}$.$\ve

Vedclass · Accepted Answer

$9$

Vedclass · Answer

(C) આપેલ સદિશો $\vec{a}=(1, -\alpha, \beta)$,$\vec{b}=(3, \beta, -\alpha)$,અને $\vec{c}=(-\alpha, -2, 1)$ છે,જ્યાં $\alpha, \beta \in \mathbb{Z}$.$\vec{a} \cdot \vec{b} = -1$ પરથી:$(1)(3) + (-\alpha)(\beta) + (\beta)(-\alpha) = -1$$3 - 2\alpha\beta = -1 \Rightarrow 2\alpha\beta = 4 \Rightarrow \alpha\beta = 2$.$\vec{b} \cdot \vec{c} = 10$ પરથી:$(3)(-\alpha) + (\beta)(-2) + (-\alpha)(1) = 10$$-3\alpha - 2\beta - \alpha = 10 \Rightarrow -4\alpha - 2\beta = 10 \Rightarrow 2\alpha + \beta = -5$.$\alpha\beta = 2$ અને $\beta = -5 - 2\alpha$ હોવાથી,આપણે $\beta$ ની કિંમત મૂકીએ:$\alpha(-5 - 2\alpha) = 2 \Rightarrow -5\alpha - 2\alpha^2 = 2 \Rightarrow 2\alpha^2 + 5\alpha + 2 = 0$.$(2\alpha + 1)(\alpha + 2) = 0$. $\alpha$ પૂર્ણાંક હોવાથી,$\alpha = -2$.તેથી $\beta = -5 - 2(-2) = -5 + 4 = -1$.હવે,અદિશ ત્રિગુણિત ગુણાકાર $[\vec{a} \vec{b} \vec{c}] = \begin{vmatrix} 1 & 2 & -1 \ 3 & -1 & 2 \ 2 & -2 & 1 \end{vmatrix}$ શોધો.$= 1(-1 + 4) - 2(3 - 4) - 1(-6 + 2)$$= 1(3) - 2(-1) - 1(-4) = 3 + 2 + 4 = 9$.

Similar Questions

જો $\overline{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ અને $\overline{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$ હોય,તો $(2 \bar{a}-\bar{b}) \cdot [(\bar{a} \times \bar{b}) \times (\bar{a}+2 \bar{b})] = $ ની કિંમત શોધો.

જો $\alpha = 2i + 3j - k$,$\beta = -i + 2j - 4k$ અને $\gamma = i + j + k$ હોય,તો $(\alpha \times \beta) \cdot (\alpha \times \gamma)$ ની કિંમત શોધો.

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