Question

Vectors A, B, C and D have the following components:

Answer 1

Answer:

The magnitude of the resultant is approximately 5.391 units.

Explanation:

We notice that vectors have the following characteristics: A magnitude and direction. That is:

$\vec v = r\cdot (\cos \theta\,\hat{i}+\sin \theta \,\hat{j})$

Where:

$r$ - Magnitude, dimensionless.

$\theta$ - Direction, measured in radians.

This expression can be rearranged as:

$\vec v = r\cdot \cos \theta \,\hat{i}+r\cdot \sin \theta \,j$

$\vec v = v_{x}\,\hat {i}+v_{y}\,\hat{j}$

Where:

$v_{x} = r\cdot \cos \theta$

$v_{y} = r\cdot \sin \theta$

If we know a set of vectors of the form $\vec {v} = v_{x}\,\hat{i}+v_{y}\,\hat{j}$, the resultant vector ($\vec {R}$) is the vectorial sum of every vector in the set:

$\vec{R} = \Sigma_{i=1}^{n} v_{i,x}\,\hat{i}+\Sigma_{i=1}^{n}v_{i,y} \,\hat{j}$

If we know that $\vec{A} = 2.5\,\hat{i}+4.3\,\hat{j}$, $\vec {B} = 6.1\,\hat{i}-2.1\,\hat{j}$, $\vec C = -3.6\,\hat{i}+1.0\,\hat{j}$ and $\vec{D} = -1.5\,\hat{i}-7.3\,\hat{j}$, the resultant vector is:

$\vec {R} = \vec {A}+\vec {B} + \vec {C} + \vec {D}$

$\vec {R}=(2.5+6.1-3.6-1.5)\,\hat{i} + (4.3-2.1+1.0-7.3)\,\hat{j}$

$\vec {R} = 3.5\,\hat{i} -4.1\,\hat{j}$

And lastly, we obtain the magnitude of the resultant by Pythagoras' Theorem:

$\|\vec R\| = \sqrt{3.5^{2}+(-4.1)^{2}}$

$\|\vec {R}\| \approx 5.391$

The magnitude of the resultant is approximately 5.391 units.