The value of alpha,so that the volume of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The volume $V$ of a parallelepiped formed by vectors $\vec{a}, \vec{b}, \vec{c}$ is given by the scalar triple product $|[\vec{a} \vec{b} \vec{c}]

Vedclass · Accepted Answer

$\frac{-1}{\sqrt{3}}$

Vedclass · Answer

(A) The volume $V$ of a parallelepiped formed by vectors $\vec{a}, \vec{b}, \vec{c}$ is given by the scalar triple product $|[\vec{a} \vec{b} \vec{c}]|$.$V = \left| \begin{vmatrix} 1 & \alpha & 1 \ 0 & 1 & \alpha \ \alpha & 0 & 1 \end{vmatrix} ight| = |1(1-0) - \alpha(0-\alpha^2) + 1(0-\alpha)| = |1 + \alpha^3 - \alpha|$.Let $f(\alpha) = 1 + \alpha^3 - \alpha$.To find the maximum,we differentiate $f(\alpha)$ with respect to $\alpha$:$f'(\alpha) = 3\alpha^2 - 1$.Setting $f'(\alpha) = 0$,we get $3\alpha^2 = 1$,which implies $\alpha = \pm \frac{1}{\sqrt{3}}$.Now,we check the second derivative: $f''(\alpha) = 6\alpha$.For $\alpha = \frac{1}{\sqrt{3}}$,$f''(\alpha) = 6(\frac{1}{\sqrt{3}}) > 0$ (local minimum).For $\alpha = -\frac{1}{\sqrt{3}}$,$f''(\alpha) = 6(-\frac{1}{\sqrt{3}}) Thus,the volume is maximum at $\alpha = -\frac{1}{\sqrt{3}}$.

The value of $\alpha$,so that the volume of the parallelepiped formed by $\hat{i}+\alpha \hat{j}+\hat{k}$,$\hat{j}+\alpha \hat{k}$,and $\alpha \hat{i}+\hat{k}$ becomes maximum,is

Similar Questions

$[i, k, j] + [k, j, i] + [j, k, i]$

If the vectors $\bar{a}=\hat{\imath}-2 \hat{\jmath}+\hat{k}$,$\bar{b}=2 \hat{\imath}-5 \hat{\jmath}+p \hat{k}$ and $\bar{c}=5 \hat{\imath}-9 \hat{\jmath}+4 \hat{k}$ are coplanar,then the value of $p$ is

$[b \times c, c \times a, a \times b]$ is equal to

For what value of $a$ is the volume of the parallelepiped formed by the vectors $\hat{i} + a\hat{j} + \hat{k}$,$\hat{j} + a\hat{k}$,and $a\hat{i} + \hat{k}$ minimum?

The value of $a$ for which the volume of the parallelepiped formed by $\hat{i} + a \hat{j} + \hat{k}$,$\hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum is

Vedclass Test Series

Exam Paper Generator

Online Exam Module