You can solve geometric problems involving vectors in 3D with the same theory as you have previously used for 2D vectors.
Geometric problems
Finding position vectors, parallel and scalar multiples of vectors and the magnitude of a vector can all help solve geometric problems with 3D vectors. Unlike vectors in 2D, it is not easy to accurately plot the coordinates and their associated vectors by hand, however a rough sketch can help visualise the problem given.
Example 1
Three points form a triangle with the following coordinates: A(−1,6,7), B(1,9,5) and C(−1,12,7).
a) Show that the triangle is isosceles.
b) Find the area of the triangle.
c) Find the coordinates of point D, so that ABCD forms a rhombus.
a) An isosceles triangle has two sides of equal length. Find the magnitude of the vectors which form the sides of the triangle AB, AC and BC.
AB=OB−OA=195−−167=23−2
AC=OC−OA=−1127−−167=060
BC=OC−OB=−1127−195=−232
∣AB∣=22+32+(−2)2=17
∣AC∣=02+62+02=6
∣BC∣=(−2)2+32+22=17
Since ∣AB∣=∣BC∣, triangle ABC is isosceles.
b) Now that the equal sides are known, a sketch can help visualise the triangle to help find the area:
Find the perpendicular height using Pythagoras' theorem:
h=(17)2−32=22
Calculate the area
A=21bh=21×6×22=62
c) Add point D to the sketch.
From the sketch, it can be seen that for ABCD to form a rhombus, AD=BC and AB=DC. Use this to find the position vector of D:
OD=OA+AD=OA+BC=−167+−232=−399
Therefore, D has coordinates (−3,9,9).
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Position vectors and displacement vectors
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Solving geometric problems
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FAQs - Frequently Asked Questions
What is a 3D vector?
A 3D vector is a vector which has magnitude and direction and occupies 3D space. It has x, y and z components.
How do you draw 3D vectors to solve geometric problems?
Unlike vectors in 2D, it is not easy to accurately plot the coordinates and their associated vectors by hand, however a rough sketch can help visualise the problem given.
How do you solve 3D vector problems?
The same theory for 2D vectors can be applied to 3D vectors in order to solve geometric problems.