# Determine the projection of the force P=10i-8j+14k lb on the directed line which originates at point (2,-5,3) and passes through the point (5,2,-4)

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We have to determine the projection of the force vector P=10i-8j+14k lb on the line originating from (2,-5,3) and going towards the point (5,2,-4)

The length of the line between (2,-5,3) and (5,2,-4) is sqrt[ (5-2)^2+(2+5)^2+(-4-3)^2]

= sqrt 107.

The unit vector along the line joinging the two points is:

(5-2)i /sqrt 107 + (2+5)j/sqrt 107 + (-4-3)k /sqrt 107

=> 3/ sqrt 107 i + 7 / sqrt 107 j - 7 / sqrt 107 k

=> 0.29 i + .67 j - .67 k

The projection of the the vector P=10i-8j+14k on the line is the dot product between the two

=> (10i - 8j + 14k) * (0.29i + 0.67j - 0.67k)

=> 2.9 - 5.42 - 9.48

=> -12

**Therefore the dot product is equal to -12 **

**Therefore the projection of P on the required line is -12.**

We'll determine the unit vector along the line L:

vL = (5-2)i /sqrt[(5-2)^2+(2+5)^2+(-4-3)^2]+(2+5)j/sqrt107+(-4-3)k/sqrt107

vL = 0.29i + 0.67j - 0.67k

Now, we'll determine tye projection of the force P on the line L:

P*vL = (10i-8j+14k)*(0.29i + 0.67j - 0.67k)

P*vL = 2.9 - 5.42 - 9.48

**P*vL = -12 lb**

**The negative result indicates that the projection has the opposite direction to the direction of the line L.**