003 Components of a 3D force with given distances

Problem 003
Which of the following correctly defines the 500 N force that passes from A(4, 0, 3) to B(0, 6, 0)?
A. 256i - 384j + 192k N
B. -256i + 384j - 192k N
C. -384i + 192j - 256k N
D. 384i - 192j + 256k N

Force through A and B of magnitude 500 Newton

Solution 003

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From the figure
${\bf r}_{AB} = -4{\bf i} + 6{\bf j} - 3{\bf k} \, \text{ m}$

Unit vector from A to B:

${\bf \lambda}_{AB} = \dfrac{{\bf r}_{AB}}{r_{AB}}$

${\bf \lambda}_{AB} = \dfrac{-4{\bf i} + 6{\bf j} - 3{\bf k}}{\sqrt{(-4)^2 + 6^2 + (-3)^2}}$

${\bf \lambda}_{AB} = -0.5121{\bf i} + 0.7682{\bf j} - 0.3841{\bf k}$

Rectangular representation of F:

${\bf F} = F \, {\bf \lambda}_{AB}$

${\bf F} = 500(-0.5121{\bf i} + 0.7682{\bf j} - 0.3841{\bf k})$

${\bf F} = -256{\bf i} + 384{\bf j} - 192{\bf k} \, \text{ N}$

Answer: B

Calculator Technique

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May work in the following CASIO models: fx-570ES, fx-570ES Plus, fx-115ES, fx-115ES Plus, fx-991ES, and fx-991ES Plus

Use VECTOR mode: [MODE] → 8:VECTOR
This mode is made primarily for vector quantities, thus, handling forces in 3D is straightforward.

Enter position vector:
[MODE] → 8:VECTOR → 1:VctA → 1:3
r = VctA = [ -4 6 -3 ]

Solve for vector F: AC
F = 500[×]([SHIFT] → [5 VECTOR] → 3:VctA ÷ [SHIFT] → [hyp Abs]( → [SHIFT] → [5 VECTOR] → 3:VctA) ) =

Calculator display: 500×(VctA÷Abs(VctA))

F = [ -256 384 -192 ] answer

Note: the unit vector λ is:
λ = [SHIFT] → [5 VECTOR] → 3:VctA ÷ [SHIFT] → [hyp Abs]( → [SHIFT] → [5 VECTOR] → 3:VctA)

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