Fill in the blanks. The vectors a =3 i -2 j +2 k and b =- i -2 k are the adjacent sides of a parallelogram. The angle between its diagonals is _________.

The vectors a =3 i -2 j +2 k and b =- i -2 k are the adjacent sides of a parallelogram. The angle between its diagonals is = 4 .Solution: We have, a =3 i -2 j +2 k and b =- i -2 k a + b =2 i -2 j and a - b =4 i -2 j +4 k Now, let is the acute angle between the diagonals a + b and a - b . = ( a + b ) ( a - b ) |( a + b )||( a - b )| = (2 i -2 j ) (4 i -2 j +4 k ) 816+4+16 =8+4 22 6 =1 2 = 4 [ 4 =1 2 ]

Fill in the blanks. The vectors a =3 i -2 j +2 k and b =- i -2 k …

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Fill in the blanks.
The vectors $\vec{\text{a}}=3\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}}=-\hat{\text{i}}-2\hat{\text{k}}$ are the adjacent sides of a parallelogram. The angle between its diagonals is _________.

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The vectors $\vec{\text{a}}=3\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}}=-\hat{\text{i}}-2\hat{\text{k}}$ are the adjacent sides of a parallelogram. The angle between its diagonals is $\theta=\frac{\pi}{4}.$Solution:
We have, $\vec{\text{a}}=3\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}}=-\hat{\text{i}}-2\hat{\text{k}}$ $\therefore\vec{\text{a}}+\vec{\text{b}}=2\hat{\text{i}}-2\hat{\text{j}}$ and $\vec{\text{a}}-\vec{\text{b}}=4\hat{\text{i}}-2\hat{\text{j}}+4\vec{\text{k}}$ Now, let $\theta$ is the acute angle between the diagonals $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}}.$ $\therefore\cos\theta=\frac{(\vec{\text{a}}+\vec{\text{b}})\cdot(\vec{\text{a}}-\vec{\text{b}})}{|(\vec{\text{a}}+\vec{\text{b}})||(\vec{\text{a}}-\vec{\text{b}})|}$ $=\frac{(2\vec{\text{i}}-2\vec{\text{j}})\cdot(4\vec{\text{i}}-2\vec{\text{j}}+4\vec{\text{k}})}{\sqrt{8}\sqrt{16+4+16}}$ $=\frac{8+4}{2\sqrt{2}\cdot6}=\frac{1}{\sqrt{2}}$ $\therefore\theta=\frac{\pi}{4}$ $\Big[\because\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}\Big]$

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