Each of the four forces acting at E has a magnitude of 28 kN. Express each force as a Cartesian vector and determine the resultant force.

#### Solution:

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We will first write the locations of points A, B, C, D, and E in Cartesian vector form.

Using the image, the locations of the points are:

B:(6i+4j+0k) m

C:(-6i+4j+0k) m

D:(-6i-4j+0k) m

E:(0i+0j+12k) m

We will now write position vectors for points from E to A, E to B, E to C, and E to D.

r_{EA}\,=\,\left\{6i-4j-12k\right\} m

r_{EB}\,=\,\left\{(6-0)i+(4-0)j+(0-12)k\right\} m

r_{EB}\,=\,\left\{6i+4j-12k\right\} m

r_{EC}\,=\,\left\{(-6-0)i+(4-0)j+(0-12)k\right\} m

r_{EC}\,=\,\left\{-6i+4j-12k\right\} m

r_{ED}\,=\,\left\{(-6-0)i+(-4-0)j+(0-12)k\right\} m

r_{ED}\,=\,\left\{-6i-4j-12k\right\} m

Next we will find the magnitude of each position vector.

magnitude of r_{EB}\,=\,\sqrt{(6)^2+(4)^2+(-12)^2}\,=\,14 m

magnitude of r_{EC}\,=\,\sqrt{(-6)^2+(4)^2+(-12)^2}\,=\,14 m

magnitude of r_{ED}\,=\,\sqrt{(-6)^2+(-4)^2+(-12)^2}\,=\,14 m

We can now write the unit vector for each position vector.

u_{EB}\,=\,\left(\dfrac{6}{14}i\,+\,\dfrac{4}{14}j\,-\,\dfrac{12}{14}k\right)

u_{EC}\,=\,\left(-\dfrac{6}{14}i\,+\,\dfrac{4}{14}j\,-\,\dfrac{12}{14}k\right)

u_{ED}\,=\,\left(-\dfrac{6}{14}i\,-\,\dfrac{4}{14}j\,-\,\dfrac{12}{14}k\right)

Let us now express each force in Cartesian vector form. Remember that each force has a magnitude of 28 kN.

F_{EA}\,=\,\left\{12i-8j-24k\right\} kN

F_{EB}\,=\,28\left(\dfrac{6}{14}i\,+\,\dfrac{4}{14}j\,-\,\dfrac{12}{14}k\right)

F_{EB}\,=\,\left\{12i+8j-24k\right\} kN

F_{EC}\,=\,28\left(-\dfrac{6}{14}i\,+\,\dfrac{4}{14}j\,-\,\dfrac{12}{14}k\right)

F_{EC}\,=\,\left\{-12i+8j-24k\right\} kN

F_{ED}\,=\,28\left(-\dfrac{6}{14}i\,-\,\dfrac{4}{14}j\,-\,\dfrac{12}{14}k\right)

F_{ED}\,=\,\left\{-12i-8j-24k\right\} kN

The resultant force is equal to each corresponding coordinate of the forces added together.

F_R\,=\,\left\{(12+12-12-12)i+(-8+8+8-8)j+(-24-24-24-24)k\right\} kN

F_R\,=\,\left\{0i+0j-96k\right\} kN

#### Final Answers:

F_{EB}\,=\,\left\{12i+8j-24k\right\} kN

F_{EC}\,=\,\left\{-12i+8j-24k\right\} kN

F_{ED}\,=\,\left\{-12i-8j-24k\right\} kN

F_R\,=\,\left\{0i+0j-96k\right\} kN