If |vec{a}| = 2sqrt{2},|vec{b}| = 3 and the a | vedclass

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

Question

(C) Let the sides of the parallelogram be $\vec{u} = 5\vec{a} + 2\vec{b}$ and $\vec{v} = \vec{a} - 3\vec{b}$.The diagonals of the parallelogram are gi

Vedclass · Accepted Answer

$\sqrt{593}$

Vedclass · Answer

(C) Let the sides of the parallelogram be $\vec{u} = 5\vec{a} + 2\vec{b}$ and $\vec{v} = \vec{a} - 3\vec{b}$.The diagonals of the parallelogram are given by $\vec{d_1} = \vec{u} + \vec{v}$ and $\vec{d_2} = \vec{u} - \vec{v}$.$\vec{d_1} = (5\vec{a} + 2\vec{b}) + (\vec{a} - 3\vec{b}) = 6\vec{a} - \vec{b}$.$\vec{d_2} = (5\vec{a} + 2\vec{b}) - (\vec{a} - 3\vec{b}) = 4\vec{a} + 5\vec{b}$.Given $|\vec{a}| = 2\sqrt{2}$,$|\vec{b}| = 3$,and $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos(\frac{\pi}{4}) = (2\sqrt{2})(3)(\frac{1}{\sqrt{2}}) = 6$.$|\vec{d_1}|^2 = |6\vec{a} - \vec{b}|^2 = 36|\vec{a}|^2 + |\vec{b}|^2 - 12(\vec{a} \cdot \vec{b}) = 36(8) + 9 - 12(6) = 288 + 9 - 72 = 225$.$|\vec{d_1}| = \sqrt{225} = 15$.$|\vec{d_2}|^2 = |4\vec{a} + 5\vec{b}|^2 = 16|\vec{a}|^2 + 25|\vec{b}|^2 + 40(\vec{a} \cdot \vec{b}) = 16(8) + 25(9) + 40(6) = 128 + 225 + 240 = 593$.$|\vec{d_2}| = \sqrt{593}$.Since $\sqrt{593} > 15$,the length of the longer diagonal is $\sqrt{593}$.

If $|\vec{a}| = 2\sqrt{2}$,$|\vec{b}| = 3$ and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$,find the length of the longer diagonal of the parallelogram whose sides are $5\vec{a} + 2\vec{b}$ and $\vec{a} - 3\vec{b}$.

Similar Questions

Let $\vec a = \hat i - \hat j,$ $\vec b = \hat i + \hat j + \hat k$ and $\vec c$ be a vector such that $\vec a \times \vec c + \vec b = 0$ and $\vec a \cdot \vec c = 4$,then ${\left| {\vec c} \right|^2}$ is equal to

If $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are orthogonal,then the value of $\lambda$ is:

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$. Then such a vector $\hat{c}$ is :

If $a$ and $b$ are unit vectors such that $a+b$ is also a unit vector,then the angle between $a$ and $b$ is . . . . . . (in $^{\circ}$)

If the force $\overrightarrow{F} = \hat{i} + 2\hat{j} + 3\hat{k}$ moves a particle from position $\vec{r_1} = \hat{i} + \hat{j} - \hat{k}$ to $\vec{r_2} = 2\hat{i} - \hat{j} + \hat{k},$ then the work done is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module