Question

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 + B⃗
A⃗
Find the magnitude of the resultant vector A⃗ + B⃗ .

Answer 1

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}`