Vectors A, B, C and D have the following components: X components (A, B, C and D respectively): +2.5 units, +6.1 units, -3.6 units and

Question

Vectors A, B, C and D have the following components:
X components (A, B, C and D respectively): +2.5 units, +6.1 units, -3.6 units and -1.5 units
Y components (A, B, C and D respectively): +4.3 units, -2.1 units, +1.0 units and -7.3 units

The magnitude of the resultant is:________

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Xavia 6 months 2021-07-27T10:54:19+00:00 1 Answers 4 views 0

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    2021-07-27T10:56:02+00:00

    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.

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