$If \to A = 2 ˆ i + 3 ˆ j - 6 ˆ k and \to B = 3 ˆ i - 6 ˆ j - 5 ˆ k are the two adjacent sides of a parallelogram then the angle between the diagonals of the parallelogram is$

{sin}^{- 1} (\frac{- 21}{\sqrt{12865}})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

{cos}^{- 1} (\frac{- 21}{\sqrt{12865}})

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

90^{ο}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

0^{ο}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B ${cos}^{- 1} (\frac{- 21}{\sqrt{12865}})$
If $\to A$ and $\to B$ are the adjacent sides of a parallelogram then the diagonals of a parallelogram will be $(\to A + \to B) and (\to A - \to B)$
So we have to find angle between $(\to A + \to B) and (\to A - \to B)$
$\to A + \to B = (2 ˆ i + 3 ˆ j - 6 ˆ k) + (3 ˆ i - 6 ˆ j - 5 ˆ k) = 5 ˆ i - 3 ˆ j - 11 ˆ k \to A - \to B = (2 ˆ i + 3 ˆ j - 6 ˆ k) - (3 ˆ i - 6 ˆ j - 5 ˆ k) = - ˆ i + 9 ˆ j - ˆ k (\to A + \to B) . (\to A - \to B) = (\to A + \to B ∣ ∣ ∣ (\to A - \to B ∣ ∣ ∣ cos θ \Rightarrow cos θ = \frac{(\to A + \to B) . (\to A - \to B)}{(\to A + \to B ∣ ∣ ∣ (\to A - \to B ∣ ∣ ∣} (\to A + \to B) . (\to A - \to B) = (5 ˆ i - 3 ˆ j - 11 ˆ k) . (- ˆ i + 9 ˆ j - ˆ k) \Rightarrow (\to A + \to B) . (\to A - \to B) = - 5 - 27 + 11 = - 21 (\to A + \to B ∣ ∣ ∣ = \sqrt{5^{2} + {(- 3)}^{2} + {(- 11)}^{2}} = \sqrt{155} (\to A - \to B ∣ ∣ ∣ = \sqrt{{(- 1)}^{2} + 9^{2} + {(- 1)}^{2}} = \sqrt{83} ∴ cos θ = \frac{- 21}{\sqrt{12865}} θ = {cos}^{- 1} (\frac{- 21}{\sqrt{12865}})$

Suggest Corrections

Similar questions

If \to A = 2 ˆ i + 3 ˆ j - 6 ˆ k and \to B = 3 ˆ i - 6 ˆ j - 5 ˆ k are the two adjacent sides of a parallelogram then the angle between the diagonals of the parallelogram is

2 \hat{i} - 4 \hat{j} - 5 \hat{k} and 2 \hat{i} + 2 \hat{j} + 3 \hat{k}

2 i + 4 ¯ j - 5 ¯ ¯ ¯ k

i + 2 ¯ j + 3 ¯ ¯ ¯ k

\vec{a} = 3 \hat{i} - 2 \hat{j} + 2 \hat{k} and \vec{b} = - \hat{i} - 2 \hat{k}

4 \hat{i} - \hat{j} - 3 \hat{k} and - 2 \hat{j} + \hat{j} - 2 \hat{k}

2 \hat{i} + \hat{k} and \hat{i} + \hat{j} + \hat{k}

3 \hat{i} + 4 \hat{j} and \hat{i} + \hat{j} + \hat{k}

2 \hat{i} + 3 \hat{j} + 6 \hat{k} and 3 \hat{i} - 6 \hat{j} + 2 \hat{k}

Join BYJU'S Learning Program