A parallelogram is determined by the vectors a = (-2,5) and b = (3,2). Determined the angles between the diagonals of the parallelogram?

Question

Noah Atkinson · Accepted Answer

Using the Law of Cosines

As Stefan indicated, the quick method is to use the Law of Cosines or Dot Product. #||<-2, 5>|| = sqrt(29)# #||<3, 2>|| = sqrt(13)# Find the two diagonals first: <-2, 5> + <3, 2> = <1, 7> #||<1, 7>|| = sqrt(50) = 5sqrt(2)# <-2, 5> - <3, 2> = <-5, 3> #||<-5, 3>|| = sqrt(34)# A triangle is formed by half of the lengths of the two diagonals, and either of the lengths of the original vectors. Using the Law of Cosines... #a^2 = b^2 + c^2 - 2abcos(alpha)# In this case... #13 = 34/4 + 50/4 - 2(sqrt(34)/2)(sqrt(50)/2)cos(alpha)# #13 = 21 - (sqrt(34))(5sqrt(2)/2)cos(alpha)# #13 = 21 - 5sqrt(17)cos(alpha)# #cos(alpha) = 8/(5sqrt(17))# #alpha = arccos(8/(5sqrt(17)))# In radians #alpha ~~ 1.7227#. The other angle is #pi - alpha ~~ 1.96932# If you prefer degrees, multiply the radian angles by #180/pi#.

Caroline Aucoin · Answer

undefined To find the angles between the diagonals of the parallelogram determined by the vectors $ \mathbf{a} = (-2, 5) $ and $ \mathbf{b} = (3, 2) $, we first need to find the vectors representing the diagonals.

The diagonals of a parallelogram are formed by the sum and difference of its adjacent sides. Therefore, the vectors representing the diagonals, denoted as $ \mathbf{d_1} $ and $ \mathbf{d_2} $, can be calculated as follows:

$$ \mathbf{d_1} = \mathbf{a} + \mathbf{b} $$
$$ \mathbf{d_2} = \mathbf{a} - \mathbf{b} $$

Next, find the magnitudes of $ \mathbf{d_1} $ and $ \mathbf{d_2} $, denoted as $ |\mathbf{d_1}| $ and $ |\mathbf{d_2}| $, respectively.

After obtaining the magnitudes, we can use the dot product formula to find the angle between two vectors:

$$ \cos \theta = \frac{\mathbf{d_1} \cdot \mathbf{d_2}}{|\mathbf{d_1}| \cdot |\mathbf{d_2}|} $$

Finally, calculate the angle $ \theta $ using the inverse cosine function:

$$ \theta = \arccos \left( \frac{\mathbf{d_1} \cdot \mathbf{d_2}}{|\mathbf{d_1}| \cdot |\mathbf{d_2}|} \right) $$

Substitute the values of $ \mathbf{d_1} $, $ \mathbf{d_2} $, $ |\mathbf{d_1}| $, and $ |\mathbf{d_2}| $ into the formula to find the angle between the diagonals.