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Futher Maths Vectors Question

Hi, I'm having trouble with this question. Usually distance between a point and a plane uses a formula, but this is only in regards to one direction, so am unsure how to answer this. The answers say to do the dot product beween (13i + j - 18k) [The normal for the mineral layer] and (5i + 12j + lambda) but I don't understand why. Could someone please help?


The ground is modelled as a horizontal plane with equation z = 0
the mineral layer is modelled as part of the plane containing the points A(10, 5, –50),
B(15, 30, –45) and C(–5, 20, –60), where the units are in metres

The mining company plans to drill vertically downwards from the point (5, 12, 0) on the
ground to reach the mineral layer.

(c) Using the model, determine, in metres to 1 decimal place, the distance the mining
company will need to drill in order to reach the mineral layer.
Reply 1
Original post by a.fallowfield
Hi, I'm having trouble with this question. Usually distance between a point and a plane uses a formula, but this is only in regards to one direction, so am unsure how to answer this. The answers say to do the dot product beween (13i + j - 18k) [The normal for the mineral layer] and (5i + 12j + lambda) but I don't understand why. Could someone please help?


The ground is modelled as a horizontal plane with equation z = 0
the mineral layer is modelled as part of the plane containing the points A(10, 5, –50),
B(15, 30, –45) and C(–5, 20, –60), where the units are in metres

The mining company plans to drill vertically downwards from the point (5, 12, 0) on the
ground to reach the mineral layer.

(c) Using the model, determine, in metres to 1 decimal place, the distance the mining
company will need to drill in order to reach the mineral layer.

The line is modelled as 5i + 12j + lambda k and this intersects with the plane
n.x = d
(where d is given by subbing one of the points in) and this intersects with the plane when
n.(5i+12j+lambdak) = d
(edited 10 months ago)
Reply 2
Ah, I was still considering the vector as a point, but ofc cuz it said fixed origin we can make that a line. That makes so much more sense now, thank you!

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