Consider two vectors A⃗ and B⃗ and their re- sultant A⃗ + B⃗ . The magnitudes of the vectors A⃗ and B⃗ are, respectively, 13.9 and 5.9 and they act at 53◦ to each other. B⃗ A⃗ +…

Question

asked Feb 1, 2023 45.2k views

1 Answer

Nassif Bourguig · Answer 1 · 2023-02-05T12:07:33+0000

Use the dot product identity,

$\vec x \cdot \vec y = \|\vec x\| \|\vec y\| \cos(\theta)$

where
$\theta$ is the measure of the angle between the two vectors
$\vec x,\vec y$ . Note that

$\vec x \cdot \vec x = \|\vec\| \|\vec x\| \cos(0) = \|\vec x\|^2$

Applying the identity to
$\vec x = \vec y = \vec A + \vec B$ and using the distributive and commutative properties of the dot product, we get

$(\vec A + \vec B) \cdot (\vec A + \vec B) = (\vec A \cdot \vec A) + 2 (\vec A \cdot \vec B) + (\vec B \cdot \vec B)$

which is equivalent to

$\|\vec A + \vec B\|^2 = \|\vec A\|^2 + 2\|\vec A\|\|\vec B\| \cos(\theta) + \|\vec B\|^2$

Now, we're given
$\|\vec A\| = 13.9$ ,
$\|\vec B\| = 5.9$ , and
$\theta=53^\circ$ , so that

$\|\vec A + \vec B\| = √(13.9^2 + 2\cdot13.9\cdot5.9\cos(53^\circ) + 5.9^2) \approx \boxed{18.1}$