Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 02 (b)

MATH152 April 2013

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Question B 02 (b)
Let $A=(0,1,2)$, $B=(1,0,3)$ and $C=(1,1,4)$. Let $\Delta$ be the triangle whose vertices are A,B, and C. Prove that $\Delta$ is a right-angled triangle.

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Hint
There is a relationship between the dot product of two vectors and these vectors being orthogonal, do you know what it is?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let ${\vec {AB}}=B-A={\begin{bmatrix}1&0&3\end{bmatrix}}-{\begin{bmatrix}0&1&2\end{bmatrix}}={\begin{bmatrix}1&-1&1\end{bmatrix}}$ and let ${\vec {BC}}=C-B={\begin{bmatrix}1&1&4\end{bmatrix}}-{\begin{bmatrix}1&0&3\end{bmatrix}}={\begin{bmatrix}0&1&1\end{bmatrix}}$ as in part (a). The vector are orthogonal if their dot product is zero. Calculating this dot product yields ${\vec {AB}}\cdot {\vec {BC}}=(1)(0)+(-1)(1)+(1)(1)=0$ Thus, vectors ${\vec {BA}}$ and ${\vec {CB}}$ are orthogonal and so the triangle is right angled.

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