Use vectors to decide whether the triangle with vertices P(1, -3, -2), Q(2, 0, -4), and R(6,-2, -5) is right-angled.

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Cristiano Sears · Accepted Answer

Step 1 Recall. To show that this triangle is a right triangle, we must find any two vectors of this triangle such that the dot product is 0. This is because if the dot product is 0, the vectors (which represent the sides) are perpendicular or orthogonal which means that the angle they form is right (90∘) Finding the directional vectors. We are given P(1,-3,-2),Q(2,0,-4), and R(6,-2,-5).We can define three directional vectors that correspond to the sides of the triangle as follows: PQ→=⟨2,0,−4⟩−⟨1,−3,−2⟩ =⟨2−1,0−(−3),−4(−2)⟩ =⟨1,3,−2⟩ QR→=⟨6,−2,−5⟩−⟨2,0,−4⟩ =⟨6−2,−2−0,−5−(−4)⟩ =⟨4,−2−1⟩ PR→=⟨6,−2,−5⟩−⟨1,−3,−2⟩ =⟨6−1,−2−(−3),−5−(−2)⟩ =⟨5,1−3⟩ Finding a set of vectors whose cross product is 0.Observe that: PQ→⋅PR→=⟨1,3,−2⟩⋅⟨5,1,−3⟩ =(1)(5)+(3)(1)+(−2)(−3) =5+3+6 =14≠0 QR→⋅PR→=⟨4,−2,−1⟩⋅⟨5,1,−3⟩ =(4)(5)+(−2)(1)+(−1)(−3) =20−2+3 =21≠0 PQ→⋅QR→=⟨1,3,−2⟩⋅⟨4,−2,−1⟩ =(1)(4)+(3)(−2)+(−2)(−1) =4−6+2 =0 We find that  PQ→⋅QR→=0  which means that these vectors (sides)  are orthogonal or perpendicular.  Since this is the case, this triangle mus be a right triangle.

Use vectors to decide whether the triangle with vertices P(1, -3, -2), Q(2, 0, -

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